Properties

Label 1728.3.b.i.1567.10
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 1567.10
Root \(1.33544 + 0.465413i\) of defining polynomial
Character \(\chi\) \(=\) 1728.1567
Dual form 1728.3.b.i.1567.3

$q$-expansion

\(f(q)\) \(=\) \(q+3.99718i q^{5} +7.74742i q^{7} +O(q^{10})\) \(q+3.99718i q^{5} +7.74742i q^{7} +7.68970 q^{11} -1.42738i q^{13} -22.3189 q^{17} -21.9464 q^{19} +18.1656i q^{23} +9.02254 q^{25} +15.2838i q^{29} -10.2197i q^{31} -30.9678 q^{35} +59.1765i q^{37} -41.3743 q^{41} -11.0016 q^{43} -63.6933i q^{47} -11.0225 q^{49} +19.2810i q^{53} +30.7371i q^{55} +16.1799 q^{59} -27.7128i q^{61} +5.70551 q^{65} +60.6039 q^{67} -134.803i q^{71} +90.0019 q^{73} +59.5753i q^{77} -12.0122i q^{79} -164.757 q^{83} -89.2129i q^{85} -35.6688 q^{89} +11.0585 q^{91} -87.7236i q^{95} -147.070 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) 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
\(5\) 3.99718i 0.799436i 0.916638 + 0.399718i \(0.130892\pi\)
−0.916638 + 0.399718i \(0.869108\pi\)
\(6\) 0 0
\(7\) 7.74742i 1.10677i 0.832924 + 0.553387i \(0.186665\pi\)
−0.832924 + 0.553387i \(0.813335\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 7.68970 0.699063 0.349532 0.936925i \(-0.386341\pi\)
0.349532 + 0.936925i \(0.386341\pi\)
\(12\) 0 0
\(13\) − 1.42738i − 0.109799i −0.998492 0.0548994i \(-0.982516\pi\)
0.998492 0.0548994i \(-0.0174838\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −22.3189 −1.31288 −0.656440 0.754379i \(-0.727938\pi\)
−0.656440 + 0.754379i \(0.727938\pi\)
\(18\) 0 0
\(19\) −21.9464 −1.15507 −0.577536 0.816365i \(-0.695986\pi\)
−0.577536 + 0.816365i \(0.695986\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 18.1656i 0.789808i 0.918722 + 0.394904i \(0.129222\pi\)
−0.918722 + 0.394904i \(0.870778\pi\)
\(24\) 0 0
\(25\) 9.02254 0.360902
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 15.2838i 0.527027i 0.964656 + 0.263514i \(0.0848814\pi\)
−0.964656 + 0.263514i \(0.915119\pi\)
\(30\) 0 0
\(31\) − 10.2197i − 0.329668i −0.986321 0.164834i \(-0.947291\pi\)
0.986321 0.164834i \(-0.0527089\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −30.9678 −0.884796
\(36\) 0 0
\(37\) 59.1765i 1.59937i 0.600423 + 0.799683i \(0.294999\pi\)
−0.600423 + 0.799683i \(0.705001\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −41.3743 −1.00913 −0.504565 0.863374i \(-0.668347\pi\)
−0.504565 + 0.863374i \(0.668347\pi\)
\(42\) 0 0
\(43\) −11.0016 −0.255852 −0.127926 0.991784i \(-0.540832\pi\)
−0.127926 + 0.991784i \(0.540832\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 63.6933i − 1.35518i −0.735442 0.677588i \(-0.763025\pi\)
0.735442 0.677588i \(-0.236975\pi\)
\(48\) 0 0
\(49\) −11.0225 −0.224950
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 19.2810i 0.363792i 0.983318 + 0.181896i \(0.0582234\pi\)
−0.983318 + 0.181896i \(0.941777\pi\)
\(54\) 0 0
\(55\) 30.7371i 0.558857i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 16.1799 0.274236 0.137118 0.990555i \(-0.456216\pi\)
0.137118 + 0.990555i \(0.456216\pi\)
\(60\) 0 0
\(61\) − 27.7128i − 0.454308i −0.973859 0.227154i \(-0.927058\pi\)
0.973859 0.227154i \(-0.0729421\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 5.70551 0.0877771
\(66\) 0 0
\(67\) 60.6039 0.904536 0.452268 0.891882i \(-0.350615\pi\)
0.452268 + 0.891882i \(0.350615\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 134.803i − 1.89864i −0.314311 0.949320i \(-0.601773\pi\)
0.314311 0.949320i \(-0.398227\pi\)
\(72\) 0 0
\(73\) 90.0019 1.23290 0.616451 0.787393i \(-0.288570\pi\)
0.616451 + 0.787393i \(0.288570\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 59.5753i 0.773705i
\(78\) 0 0
\(79\) − 12.0122i − 0.152054i −0.997106 0.0760268i \(-0.975777\pi\)
0.997106 0.0760268i \(-0.0242234\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −164.757 −1.98503 −0.992513 0.122137i \(-0.961025\pi\)
−0.992513 + 0.122137i \(0.961025\pi\)
\(84\) 0 0
\(85\) − 89.2129i − 1.04956i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −35.6688 −0.400773 −0.200387 0.979717i \(-0.564220\pi\)
−0.200387 + 0.979717i \(0.564220\pi\)
\(90\) 0 0
\(91\) 11.0585 0.121522
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 87.7236i − 0.923407i
\(96\) 0 0
\(97\) −147.070 −1.51618 −0.758090 0.652149i \(-0.773867\pi\)
−0.758090 + 0.652149i \(0.773867\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 74.2140i − 0.734793i −0.930065 0.367396i \(-0.880249\pi\)
0.930065 0.367396i \(-0.119751\pi\)
\(102\) 0 0
\(103\) − 55.0676i − 0.534637i −0.963608 0.267319i \(-0.913862\pi\)
0.963608 0.267319i \(-0.0861376\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −32.6046 −0.304716 −0.152358 0.988325i \(-0.548687\pi\)
−0.152358 + 0.988325i \(0.548687\pi\)
\(108\) 0 0
\(109\) 29.6714i 0.272215i 0.990694 + 0.136108i \(0.0434593\pi\)
−0.990694 + 0.136108i \(0.956541\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −213.399 −1.88848 −0.944242 0.329251i \(-0.893204\pi\)
−0.944242 + 0.329251i \(0.893204\pi\)
\(114\) 0 0
\(115\) −72.6111 −0.631401
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 172.914i − 1.45306i
\(120\) 0 0
\(121\) −61.8686 −0.511311
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 135.994i 1.08795i
\(126\) 0 0
\(127\) − 64.2526i − 0.505926i −0.967476 0.252963i \(-0.918595\pi\)
0.967476 0.252963i \(-0.0814051\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −92.1030 −0.703076 −0.351538 0.936174i \(-0.614341\pi\)
−0.351538 + 0.936174i \(0.614341\pi\)
\(132\) 0 0
\(133\) − 170.028i − 1.27840i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −169.582 −1.23783 −0.618914 0.785459i \(-0.712427\pi\)
−0.618914 + 0.785459i \(0.712427\pi\)
\(138\) 0 0
\(139\) 71.6056 0.515148 0.257574 0.966259i \(-0.417077\pi\)
0.257574 + 0.966259i \(0.417077\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 10.9761i − 0.0767563i
\(144\) 0 0
\(145\) −61.0921 −0.421325
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 198.277i 1.33072i 0.746525 + 0.665358i \(0.231721\pi\)
−0.746525 + 0.665358i \(0.768279\pi\)
\(150\) 0 0
\(151\) 72.3305i 0.479010i 0.970895 + 0.239505i \(0.0769852\pi\)
−0.970895 + 0.239505i \(0.923015\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 40.8501 0.263549
\(156\) 0 0
\(157\) 124.063i 0.790208i 0.918636 + 0.395104i \(0.129291\pi\)
−0.918636 + 0.395104i \(0.870709\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −140.736 −0.874139
\(162\) 0 0
\(163\) 232.533 1.42659 0.713293 0.700866i \(-0.247203\pi\)
0.713293 + 0.700866i \(0.247203\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 43.9755i − 0.263327i −0.991294 0.131663i \(-0.957968\pi\)
0.991294 0.131663i \(-0.0420318\pi\)
\(168\) 0 0
\(169\) 166.963 0.987944
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 305.908i 1.76826i 0.467245 + 0.884128i \(0.345247\pi\)
−0.467245 + 0.884128i \(0.654753\pi\)
\(174\) 0 0
\(175\) 69.9014i 0.399437i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 2.19254 0.0122488 0.00612442 0.999981i \(-0.498051\pi\)
0.00612442 + 0.999981i \(0.498051\pi\)
\(180\) 0 0
\(181\) 132.855i 0.734003i 0.930220 + 0.367001i \(0.119616\pi\)
−0.930220 + 0.367001i \(0.880384\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −236.539 −1.27859
\(186\) 0 0
\(187\) −171.626 −0.917785
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 151.417i − 0.792759i −0.918087 0.396379i \(-0.870267\pi\)
0.918087 0.396379i \(-0.129733\pi\)
\(192\) 0 0
\(193\) 229.850 1.19093 0.595466 0.803381i \(-0.296967\pi\)
0.595466 + 0.803381i \(0.296967\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 189.195i − 0.960380i −0.877164 0.480190i \(-0.840568\pi\)
0.877164 0.480190i \(-0.159432\pi\)
\(198\) 0 0
\(199\) − 217.686i − 1.09390i −0.837166 0.546949i \(-0.815789\pi\)
0.837166 0.546949i \(-0.184211\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −118.410 −0.583300
\(204\) 0 0
\(205\) − 165.381i − 0.806735i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −168.761 −0.807468
\(210\) 0 0
\(211\) −247.120 −1.17118 −0.585591 0.810606i \(-0.699138\pi\)
−0.585591 + 0.810606i \(0.699138\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 43.9755i − 0.204537i
\(216\) 0 0
\(217\) 79.1765 0.364869
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 31.8577i 0.144153i
\(222\) 0 0
\(223\) 361.006i 1.61886i 0.587216 + 0.809430i \(0.300224\pi\)
−0.587216 + 0.809430i \(0.699776\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −21.0546 −0.0927515 −0.0463758 0.998924i \(-0.514767\pi\)
−0.0463758 + 0.998924i \(0.514767\pi\)
\(228\) 0 0
\(229\) 319.313i 1.39438i 0.716886 + 0.697191i \(0.245567\pi\)
−0.716886 + 0.697191i \(0.754433\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −197.607 −0.848098 −0.424049 0.905639i \(-0.639392\pi\)
−0.424049 + 0.905639i \(0.639392\pi\)
\(234\) 0 0
\(235\) 254.594 1.08338
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 256.325i − 1.07249i −0.844062 0.536245i \(-0.819842\pi\)
0.844062 0.536245i \(-0.180158\pi\)
\(240\) 0 0
\(241\) −375.717 −1.55899 −0.779495 0.626409i \(-0.784524\pi\)
−0.779495 + 0.626409i \(0.784524\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 44.0591i − 0.179833i
\(246\) 0 0
\(247\) 31.3259i 0.126825i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −423.810 −1.68849 −0.844243 0.535961i \(-0.819950\pi\)
−0.844243 + 0.535961i \(0.819950\pi\)
\(252\) 0 0
\(253\) 139.688i 0.552126i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 205.754 0.800601 0.400300 0.916384i \(-0.368906\pi\)
0.400300 + 0.916384i \(0.368906\pi\)
\(258\) 0 0
\(259\) −458.466 −1.77014
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 75.9941i 0.288951i 0.989508 + 0.144476i \(0.0461495\pi\)
−0.989508 + 0.144476i \(0.953851\pi\)
\(264\) 0 0
\(265\) −77.0695 −0.290828
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 327.627i 1.21794i 0.793192 + 0.608971i \(0.208418\pi\)
−0.793192 + 0.608971i \(0.791582\pi\)
\(270\) 0 0
\(271\) − 277.780i − 1.02502i −0.858681 0.512510i \(-0.828716\pi\)
0.858681 0.512510i \(-0.171284\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 69.3806 0.252293
\(276\) 0 0
\(277\) − 92.9929i − 0.335714i −0.985811 0.167857i \(-0.946315\pi\)
0.985811 0.167857i \(-0.0536847\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 190.784 0.678947 0.339474 0.940616i \(-0.389751\pi\)
0.339474 + 0.940616i \(0.389751\pi\)
\(282\) 0 0
\(283\) −153.511 −0.542441 −0.271220 0.962517i \(-0.587427\pi\)
−0.271220 + 0.962517i \(0.587427\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 320.544i − 1.11688i
\(288\) 0 0
\(289\) 209.135 0.723651
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 550.059i − 1.87733i −0.344825 0.938667i \(-0.612062\pi\)
0.344825 0.938667i \(-0.387938\pi\)
\(294\) 0 0
\(295\) 64.6741i 0.219234i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 25.9293 0.0867200
\(300\) 0 0
\(301\) − 85.2343i − 0.283171i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 110.773 0.363191
\(306\) 0 0
\(307\) 403.571 1.31456 0.657282 0.753645i \(-0.271706\pi\)
0.657282 + 0.753645i \(0.271706\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 328.988i − 1.05784i −0.848672 0.528919i \(-0.822598\pi\)
0.848672 0.528919i \(-0.177402\pi\)
\(312\) 0 0
\(313\) −168.133 −0.537167 −0.268584 0.963256i \(-0.586556\pi\)
−0.268584 + 0.963256i \(0.586556\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 171.437i 0.540809i 0.962747 + 0.270405i \(0.0871575\pi\)
−0.962747 + 0.270405i \(0.912843\pi\)
\(318\) 0 0
\(319\) 117.528i 0.368425i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 489.820 1.51647
\(324\) 0 0
\(325\) − 12.8786i − 0.0396266i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 493.459 1.49987
\(330\) 0 0
\(331\) −520.075 −1.57122 −0.785612 0.618720i \(-0.787652\pi\)
−0.785612 + 0.618720i \(0.787652\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 242.245i 0.723119i
\(336\) 0 0
\(337\) 411.987 1.22251 0.611257 0.791432i \(-0.290664\pi\)
0.611257 + 0.791432i \(0.290664\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 78.5866i − 0.230459i
\(342\) 0 0
\(343\) 294.227i 0.857806i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −309.229 −0.891151 −0.445575 0.895244i \(-0.647001\pi\)
−0.445575 + 0.895244i \(0.647001\pi\)
\(348\) 0 0
\(349\) − 609.516i − 1.74646i −0.487306 0.873231i \(-0.662020\pi\)
0.487306 0.873231i \(-0.337980\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −376.662 −1.06703 −0.533515 0.845791i \(-0.679129\pi\)
−0.533515 + 0.845791i \(0.679129\pi\)
\(354\) 0 0
\(355\) 538.834 1.51784
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 186.196i 0.518652i 0.965790 + 0.259326i \(0.0835003\pi\)
−0.965790 + 0.259326i \(0.916500\pi\)
\(360\) 0 0
\(361\) 120.643 0.334192
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 359.754i 0.985627i
\(366\) 0 0
\(367\) − 89.1514i − 0.242919i −0.992596 0.121460i \(-0.961243\pi\)
0.992596 0.121460i \(-0.0387575\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −149.378 −0.402636
\(372\) 0 0
\(373\) 103.487i 0.277444i 0.990331 + 0.138722i \(0.0442995\pi\)
−0.990331 + 0.138722i \(0.955701\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 21.8158 0.0578669
\(378\) 0 0
\(379\) 636.105 1.67838 0.839188 0.543841i \(-0.183031\pi\)
0.839188 + 0.543841i \(0.183031\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 617.514i 1.61231i 0.591707 + 0.806153i \(0.298454\pi\)
−0.591707 + 0.806153i \(0.701546\pi\)
\(384\) 0 0
\(385\) −238.133 −0.618528
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 674.534i − 1.73402i −0.498290 0.867010i \(-0.666039\pi\)
0.498290 0.867010i \(-0.333961\pi\)
\(390\) 0 0
\(391\) − 405.437i − 1.03692i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 48.0151 0.121557
\(396\) 0 0
\(397\) 758.512i 1.91061i 0.295622 + 0.955305i \(0.404473\pi\)
−0.295622 + 0.955305i \(0.595527\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 314.381 0.783993 0.391997 0.919967i \(-0.371784\pi\)
0.391997 + 0.919967i \(0.371784\pi\)
\(402\) 0 0
\(403\) −14.5875 −0.0361972
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 455.050i 1.11806i
\(408\) 0 0
\(409\) −591.015 −1.44503 −0.722513 0.691358i \(-0.757013\pi\)
−0.722513 + 0.691358i \(0.757013\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 125.353i 0.303517i
\(414\) 0 0
\(415\) − 658.564i − 1.58690i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 371.686 0.887078 0.443539 0.896255i \(-0.353723\pi\)
0.443539 + 0.896255i \(0.353723\pi\)
\(420\) 0 0
\(421\) − 660.963i − 1.56998i −0.619507 0.784991i \(-0.712667\pi\)
0.619507 0.784991i \(-0.287333\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −201.374 −0.473820
\(426\) 0 0
\(427\) 214.703 0.502817
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 376.295i 0.873074i 0.899686 + 0.436537i \(0.143795\pi\)
−0.899686 + 0.436537i \(0.856205\pi\)
\(432\) 0 0
\(433\) 301.156 0.695510 0.347755 0.937585i \(-0.386944\pi\)
0.347755 + 0.937585i \(0.386944\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 398.669i − 0.912285i
\(438\) 0 0
\(439\) 463.627i 1.05610i 0.849214 + 0.528049i \(0.177076\pi\)
−0.849214 + 0.528049i \(0.822924\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 655.755 1.48026 0.740130 0.672464i \(-0.234764\pi\)
0.740130 + 0.672464i \(0.234764\pi\)
\(444\) 0 0
\(445\) − 142.575i − 0.320393i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 167.436 0.372909 0.186455 0.982464i \(-0.440300\pi\)
0.186455 + 0.982464i \(0.440300\pi\)
\(450\) 0 0
\(451\) −318.156 −0.705446
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 44.2030i 0.0971495i
\(456\) 0 0
\(457\) −527.115 −1.15342 −0.576712 0.816948i \(-0.695664\pi\)
−0.576712 + 0.816948i \(0.695664\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 395.196i 0.857259i 0.903480 + 0.428629i \(0.141003\pi\)
−0.903480 + 0.428629i \(0.858997\pi\)
\(462\) 0 0
\(463\) − 159.767i − 0.345070i −0.985003 0.172535i \(-0.944804\pi\)
0.985003 0.172535i \(-0.0551958\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 275.233 0.589363 0.294682 0.955595i \(-0.404786\pi\)
0.294682 + 0.955595i \(0.404786\pi\)
\(468\) 0 0
\(469\) 469.524i 1.00112i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −84.5993 −0.178857
\(474\) 0 0
\(475\) −198.012 −0.416867
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 551.742i 1.15186i 0.817498 + 0.575931i \(0.195360\pi\)
−0.817498 + 0.575931i \(0.804640\pi\)
\(480\) 0 0
\(481\) 84.4676 0.175608
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 587.864i − 1.21209i
\(486\) 0 0
\(487\) − 219.363i − 0.450437i −0.974308 0.225218i \(-0.927690\pi\)
0.974308 0.225218i \(-0.0723095\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 346.740 0.706191 0.353095 0.935587i \(-0.385129\pi\)
0.353095 + 0.935587i \(0.385129\pi\)
\(492\) 0 0
\(493\) − 341.118i − 0.691923i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1044.38 2.10137
\(498\) 0 0
\(499\) 101.861 0.204130 0.102065 0.994778i \(-0.467455\pi\)
0.102065 + 0.994778i \(0.467455\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 752.544i 1.49611i 0.663635 + 0.748056i \(0.269013\pi\)
−0.663635 + 0.748056i \(0.730987\pi\)
\(504\) 0 0
\(505\) 296.647 0.587420
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 339.596i 0.667182i 0.942718 + 0.333591i \(0.108260\pi\)
−0.942718 + 0.333591i \(0.891740\pi\)
\(510\) 0 0
\(511\) 697.283i 1.36455i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 220.115 0.427408
\(516\) 0 0
\(517\) − 489.782i − 0.947354i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 683.717 1.31232 0.656158 0.754623i \(-0.272180\pi\)
0.656158 + 0.754623i \(0.272180\pi\)
\(522\) 0 0
\(523\) −112.671 −0.215433 −0.107716 0.994182i \(-0.534354\pi\)
−0.107716 + 0.994182i \(0.534354\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 228.093i 0.432815i
\(528\) 0 0
\(529\) 199.011 0.376203
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 59.0571i 0.110801i
\(534\) 0 0
\(535\) − 130.327i − 0.243601i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −84.7600 −0.157254
\(540\) 0 0
\(541\) 752.788i 1.39147i 0.718296 + 0.695737i \(0.244922\pi\)
−0.718296 + 0.695737i \(0.755078\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −118.602 −0.217619
\(546\) 0 0
\(547\) −892.328 −1.63131 −0.815657 0.578536i \(-0.803624\pi\)
−0.815657 + 0.578536i \(0.803624\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) − 335.424i − 0.608754i
\(552\) 0 0
\(553\) 93.0638 0.168289
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 105.929i − 0.190177i −0.995469 0.0950887i \(-0.969687\pi\)
0.995469 0.0950887i \(-0.0303135\pi\)
\(558\) 0 0
\(559\) 15.7036i 0.0280922i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −221.818 −0.393993 −0.196997 0.980404i \(-0.563119\pi\)
−0.196997 + 0.980404i \(0.563119\pi\)
\(564\) 0 0
\(565\) − 852.994i − 1.50972i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 604.024 1.06155 0.530777 0.847511i \(-0.321900\pi\)
0.530777 + 0.847511i \(0.321900\pi\)
\(570\) 0 0
\(571\) 470.074 0.823248 0.411624 0.911354i \(-0.364962\pi\)
0.411624 + 0.911354i \(0.364962\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 163.900i 0.285043i
\(576\) 0 0
\(577\) 484.209 0.839183 0.419592 0.907713i \(-0.362173\pi\)
0.419592 + 0.907713i \(0.362173\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 1276.44i − 2.19698i
\(582\) 0 0
\(583\) 148.265i 0.254314i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 361.426 0.615717 0.307858 0.951432i \(-0.400388\pi\)
0.307858 + 0.951432i \(0.400388\pi\)
\(588\) 0 0
\(589\) 224.286i 0.380791i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 277.115 0.467310 0.233655 0.972320i \(-0.424931\pi\)
0.233655 + 0.972320i \(0.424931\pi\)
\(594\) 0 0
\(595\) 691.170 1.16163
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 654.345i 1.09240i 0.837656 + 0.546198i \(0.183925\pi\)
−0.837656 + 0.546198i \(0.816075\pi\)
\(600\) 0 0
\(601\) −138.002 −0.229620 −0.114810 0.993387i \(-0.536626\pi\)
−0.114810 + 0.993387i \(0.536626\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 247.300i − 0.408760i
\(606\) 0 0
\(607\) 569.498i 0.938217i 0.883140 + 0.469109i \(0.155425\pi\)
−0.883140 + 0.469109i \(0.844575\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −90.9148 −0.148797
\(612\) 0 0
\(613\) 1132.55i 1.84756i 0.382925 + 0.923779i \(0.374917\pi\)
−0.382925 + 0.923779i \(0.625083\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 651.926 1.05661 0.528303 0.849056i \(-0.322829\pi\)
0.528303 + 0.849056i \(0.322829\pi\)
\(618\) 0 0
\(619\) −691.283 −1.11677 −0.558387 0.829580i \(-0.688580\pi\)
−0.558387 + 0.829580i \(0.688580\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 276.341i − 0.443566i
\(624\) 0 0
\(625\) −318.030 −0.508848
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 1320.76i − 2.09977i
\(630\) 0 0
\(631\) 1008.39i 1.59808i 0.601280 + 0.799038i \(0.294658\pi\)
−0.601280 + 0.799038i \(0.705342\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 256.829 0.404455
\(636\) 0 0
\(637\) 15.7334i 0.0246992i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 609.937 0.951540 0.475770 0.879570i \(-0.342169\pi\)
0.475770 + 0.879570i \(0.342169\pi\)
\(642\) 0 0
\(643\) −453.534 −0.705341 −0.352670 0.935748i \(-0.614726\pi\)
−0.352670 + 0.935748i \(0.614726\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 422.050i 0.652319i 0.945315 + 0.326159i \(0.105755\pi\)
−0.945315 + 0.326159i \(0.894245\pi\)
\(648\) 0 0
\(649\) 124.419 0.191708
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 912.070i 1.39674i 0.715738 + 0.698369i \(0.246091\pi\)
−0.715738 + 0.698369i \(0.753909\pi\)
\(654\) 0 0
\(655\) − 368.152i − 0.562065i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −1130.79 −1.71591 −0.857957 0.513722i \(-0.828266\pi\)
−0.857957 + 0.513722i \(0.828266\pi\)
\(660\) 0 0
\(661\) 669.361i 1.01265i 0.862343 + 0.506324i \(0.168996\pi\)
−0.862343 + 0.506324i \(0.831004\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 679.632 1.02200
\(666\) 0 0
\(667\) −277.639 −0.416250
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 213.103i − 0.317590i
\(672\) 0 0
\(673\) −504.561 −0.749719 −0.374859 0.927082i \(-0.622309\pi\)
−0.374859 + 0.927082i \(0.622309\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 863.692i − 1.27576i −0.770134 0.637882i \(-0.779811\pi\)
0.770134 0.637882i \(-0.220189\pi\)
\(678\) 0 0
\(679\) − 1139.41i − 1.67807i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 845.591 1.23805 0.619027 0.785369i \(-0.287527\pi\)
0.619027 + 0.785369i \(0.287527\pi\)
\(684\) 0 0
\(685\) − 677.852i − 0.989565i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 27.5213 0.0399439
\(690\) 0 0
\(691\) −165.670 −0.239753 −0.119877 0.992789i \(-0.538250\pi\)
−0.119877 + 0.992789i \(0.538250\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 286.220i 0.411828i
\(696\) 0 0
\(697\) 923.432 1.32487
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 947.047i 1.35099i 0.737362 + 0.675497i \(0.236071\pi\)
−0.737362 + 0.675497i \(0.763929\pi\)
\(702\) 0 0
\(703\) − 1298.71i − 1.84738i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 574.967 0.813250
\(708\) 0 0
\(709\) 835.746i 1.17877i 0.807853 + 0.589383i \(0.200629\pi\)
−0.807853 + 0.589383i \(0.799371\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 185.647 0.260375
\(714\) 0 0
\(715\) 43.8737 0.0613618
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 20.1727i 0.0280566i 0.999902 + 0.0140283i \(0.00446549\pi\)
−0.999902 + 0.0140283i \(0.995535\pi\)
\(720\) 0 0
\(721\) 426.632 0.591723
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 137.899i 0.190205i
\(726\) 0 0
\(727\) − 276.178i − 0.379887i −0.981795 0.189943i \(-0.939170\pi\)
0.981795 0.189943i \(-0.0608305\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 245.545 0.335903
\(732\) 0 0
\(733\) 294.820i 0.402210i 0.979570 + 0.201105i \(0.0644533\pi\)
−0.979570 + 0.201105i \(0.935547\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 466.026 0.632328
\(738\) 0 0
\(739\) 914.711 1.23777 0.618885 0.785482i \(-0.287585\pi\)
0.618885 + 0.785482i \(0.287585\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1068.78i 1.43846i 0.694772 + 0.719230i \(0.255505\pi\)
−0.694772 + 0.719230i \(0.744495\pi\)
\(744\) 0 0
\(745\) −792.548 −1.06382
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 252.602i − 0.337252i
\(750\) 0 0
\(751\) 1147.59i 1.52809i 0.645164 + 0.764044i \(0.276789\pi\)
−0.645164 + 0.764044i \(0.723211\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −289.118 −0.382938
\(756\) 0 0
\(757\) − 91.4460i − 0.120801i −0.998174 0.0604003i \(-0.980762\pi\)
0.998174 0.0604003i \(-0.0192377\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −762.676 −1.00220 −0.501101 0.865389i \(-0.667072\pi\)
−0.501101 + 0.865389i \(0.667072\pi\)
\(762\) 0 0
\(763\) −229.877 −0.301281
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 23.0950i − 0.0301108i
\(768\) 0 0
\(769\) −214.567 −0.279021 −0.139511 0.990221i \(-0.544553\pi\)
−0.139511 + 0.990221i \(0.544553\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 459.360i 0.594257i 0.954838 + 0.297128i \(0.0960289\pi\)
−0.954838 + 0.297128i \(0.903971\pi\)
\(774\) 0 0
\(775\) − 92.2079i − 0.118978i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 908.017 1.16562
\(780\) 0 0
\(781\) − 1036.60i − 1.32727i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −495.901 −0.631721
\(786\) 0 0
\(787\) 76.5940 0.0973241 0.0486620 0.998815i \(-0.484504\pi\)
0.0486620 + 0.998815i \(0.484504\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) − 1653.29i − 2.09013i
\(792\) 0 0
\(793\) −39.5568 −0.0498825
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 273.548i 0.343223i 0.985165 + 0.171611i \(0.0548973\pi\)
−0.985165 + 0.171611i \(0.945103\pi\)
\(798\) 0 0
\(799\) 1421.57i 1.77918i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 692.087 0.861877
\(804\) 0 0
\(805\) − 562.549i − 0.698819i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −999.127 −1.23502 −0.617508 0.786565i \(-0.711858\pi\)
−0.617508 + 0.786565i \(0.711858\pi\)
\(810\) 0 0
\(811\) 1189.75 1.46702 0.733509 0.679679i \(-0.237881\pi\)
0.733509 + 0.679679i \(0.237881\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 929.478i 1.14046i
\(816\) 0 0
\(817\) 241.446 0.295528
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 985.174i − 1.19997i −0.800012 0.599984i \(-0.795174\pi\)
0.800012 0.599984i \(-0.204826\pi\)
\(822\) 0 0
\(823\) − 186.578i − 0.226705i −0.993555 0.113352i \(-0.963841\pi\)
0.993555 0.113352i \(-0.0361589\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −189.836 −0.229548 −0.114774 0.993392i \(-0.536614\pi\)
−0.114774 + 0.993392i \(0.536614\pi\)
\(828\) 0 0
\(829\) − 920.478i − 1.11035i −0.831735 0.555174i \(-0.812652\pi\)
0.831735 0.555174i \(-0.187348\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 246.011 0.295332
\(834\) 0 0
\(835\) 175.778 0.210513
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 584.287i 0.696408i 0.937419 + 0.348204i \(0.113208\pi\)
−0.937419 + 0.348204i \(0.886792\pi\)
\(840\) 0 0
\(841\) 607.406 0.722242
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 667.380i 0.789798i
\(846\) 0 0
\(847\) − 479.322i − 0.565906i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −1074.98 −1.26319
\(852\) 0 0
\(853\) − 767.535i − 0.899806i −0.893077 0.449903i \(-0.851458\pi\)
0.893077 0.449903i \(-0.148542\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −30.7621 −0.0358951 −0.0179476 0.999839i \(-0.505713\pi\)
−0.0179476 + 0.999839i \(0.505713\pi\)
\(858\) 0 0
\(859\) −795.268 −0.925807 −0.462903 0.886409i \(-0.653192\pi\)
−0.462903 + 0.886409i \(0.653192\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1107.46i 1.28327i 0.767012 + 0.641633i \(0.221743\pi\)
−0.767012 + 0.641633i \(0.778257\pi\)
\(864\) 0 0
\(865\) −1222.77 −1.41361
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 92.3704i − 0.106295i
\(870\) 0 0
\(871\) − 86.5051i − 0.0993169i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −1053.60 −1.20412
\(876\) 0 0
\(877\) 649.012i 0.740037i 0.929024 + 0.370018i \(0.120649\pi\)
−0.929024 + 0.370018i \(0.879351\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −951.863 −1.08043 −0.540217 0.841526i \(-0.681658\pi\)
−0.540217 + 0.841526i \(0.681658\pi\)
\(882\) 0 0
\(883\) −431.588 −0.488774 −0.244387 0.969678i \(-0.578587\pi\)
−0.244387 + 0.969678i \(0.578587\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 101.304i 0.114209i 0.998368 + 0.0571047i \(0.0181869\pi\)
−0.998368 + 0.0571047i \(0.981813\pi\)
\(888\) 0 0
\(889\) 497.792 0.559946
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 1397.84i 1.56533i
\(894\) 0 0
\(895\) 8.76399i 0.00979216i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 156.196 0.173744
\(900\) 0 0
\(901\) − 430.331i − 0.477615i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −531.044 −0.586789
\(906\) 0 0
\(907\) 489.763 0.539981 0.269991 0.962863i \(-0.412979\pi\)
0.269991 + 0.962863i \(0.412979\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 253.105i − 0.277832i −0.990304 0.138916i \(-0.955638\pi\)
0.990304 0.138916i \(-0.0443617\pi\)
\(912\) 0 0
\(913\) −1266.93 −1.38766
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 713.561i − 0.778147i
\(918\) 0 0
\(919\) 68.3565i 0.0743813i 0.999308 + 0.0371907i \(0.0118409\pi\)
−0.999308 + 0.0371907i \(0.988159\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −192.416 −0.208468
\(924\) 0 0
\(925\) 533.923i 0.577214i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −319.311 −0.343715 −0.171857 0.985122i \(-0.554977\pi\)
−0.171857 + 0.985122i \(0.554977\pi\)
\(930\) 0 0
\(931\) 241.905 0.259833
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) − 686.020i − 0.733711i
\(936\) 0 0
\(937\) −680.412 −0.726161 −0.363080 0.931758i \(-0.618275\pi\)
−0.363080 + 0.931758i \(0.618275\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 265.255i 0.281886i 0.990018 + 0.140943i \(0.0450134\pi\)
−0.990018 + 0.140943i \(0.954987\pi\)
\(942\) 0 0
\(943\) − 751.589i − 0.797019i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 412.117 0.435182 0.217591 0.976040i \(-0.430180\pi\)
0.217591 + 0.976040i \(0.430180\pi\)
\(948\) 0 0
\(949\) − 128.467i − 0.135371i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −149.202 −0.156561 −0.0782804 0.996931i \(-0.524943\pi\)
−0.0782804 + 0.996931i \(0.524943\pi\)
\(954\) 0 0
\(955\) 605.241 0.633760
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 1313.83i − 1.37000i
\(960\) 0 0
\(961\) 856.557 0.891319
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 918.752i 0.952074i
\(966\) 0 0
\(967\) − 91.2392i − 0.0943528i −0.998887 0.0471764i \(-0.984978\pi\)
0.998887 0.0471764i \(-0.0150223\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −117.427 −0.120934 −0.0604669 0.998170i \(-0.519259\pi\)
−0.0604669 + 0.998170i \(0.519259\pi\)
\(972\) 0 0
\(973\) 554.758i 0.570153i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −576.207 −0.589772 −0.294886 0.955532i \(-0.595282\pi\)
−0.294886 + 0.955532i \(0.595282\pi\)
\(978\) 0 0
\(979\) −274.282 −0.280166
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 1338.61i − 1.36176i −0.732395 0.680880i \(-0.761597\pi\)
0.732395 0.680880i \(-0.238403\pi\)
\(984\) 0 0
\(985\) 756.246 0.767763
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 199.851i − 0.202074i
\(990\) 0 0
\(991\) − 1519.83i − 1.53363i −0.641867 0.766816i \(-0.721840\pi\)
0.641867 0.766816i \(-0.278160\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 870.129 0.874501
\(996\) 0 0
\(997\) − 492.073i − 0.493554i −0.969072 0.246777i \(-0.920628\pi\)
0.969072 0.246777i \(-0.0793715\pi\)
\(998\) 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.1567.10 yes 12
3.2 odd 2 1728.3.b.j.1567.4 yes 12
4.3 odd 2 inner 1728.3.b.i.1567.9 yes 12
8.3 odd 2 inner 1728.3.b.i.1567.3 12
8.5 even 2 inner 1728.3.b.i.1567.4 yes 12
12.11 even 2 1728.3.b.j.1567.3 yes 12
24.5 odd 2 1728.3.b.j.1567.10 yes 12
24.11 even 2 1728.3.b.j.1567.9 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1728.3.b.i.1567.3 12 8.3 odd 2 inner
1728.3.b.i.1567.4 yes 12 8.5 even 2 inner
1728.3.b.i.1567.9 yes 12 4.3 odd 2 inner
1728.3.b.i.1567.10 yes 12 1.1 even 1 trivial
1728.3.b.j.1567.3 yes 12 12.11 even 2
1728.3.b.j.1567.4 yes 12 3.2 odd 2
1728.3.b.j.1567.9 yes 12 24.11 even 2
1728.3.b.j.1567.10 yes 12 24.5 odd 2