Properties

Label 1728.3.b.j.1567.9
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 1567.9
Root \(1.33544 + 0.465413i\) of defining polynomial
Character \(\chi\) \(=\) 1728.1567
Dual form 1728.3.b.j.1567.4

$q$-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 \(-0.813335\pi\)
0.832924 0.553387i \(-0.186665\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −7.68970 −0.699063 −0.349532 0.936925i \(-0.613659\pi\)
−0.349532 + 0.936925i \(0.613659\pi\)
\(12\) 0 0
\(13\) 1.42738i 0.109799i 0.998492 + 0.0548994i \(0.0174838\pi\)
−0.998492 + 0.0548994i \(0.982516\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 22.3189 1.31288 0.656440 0.754379i \(-0.272062\pi\)
0.656440 + 0.754379i \(0.272062\pi\)
\(18\) 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 \(0.0527089\pi\)
−0.986321 + 0.164834i \(0.947291\pi\)
\(32\) 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 \(-0.705001\pi\)
0.600423 0.799683i \(-0.294999\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 41.3743 1.00913 0.504565 0.863374i \(-0.331653\pi\)
0.504565 + 0.863374i \(0.331653\pi\)
\(42\) 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 \(-0.543784\pi\)
−0.137118 + 0.990555i \(0.543784\pi\)
\(60\) 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
\(65\) −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 \(0.0242234\pi\)
−0.997106 + 0.0760268i \(0.975777\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 164.757 1.98503 0.992513 0.122137i \(-0.0389746\pi\)
0.992513 + 0.122137i \(0.0389746\pi\)
\(84\) 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 \(-0.435780\pi\)
0.200387 + 0.979717i \(0.435780\pi\)
\(90\) 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 \(0.0861376\pi\)
−0.963608 + 0.267319i \(0.913862\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 32.6046 0.304716 0.152358 0.988325i \(-0.451313\pi\)
0.152358 + 0.988325i \(0.451313\pi\)
\(108\) 0 0
\(109\) − 29.6714i − 0.272215i −0.990694 0.136108i \(-0.956541\pi\)
0.990694 0.136108i \(-0.0434593\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 213.399 1.88848 0.944242 0.329251i \(-0.106796\pi\)
0.944242 + 0.329251i \(0.106796\pi\)
\(114\) 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 \(0.0814051\pi\)
−0.967476 + 0.252963i \(0.918595\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 92.1030 0.703076 0.351538 0.936174i \(-0.385659\pi\)
0.351538 + 0.936174i \(0.385659\pi\)
\(132\) 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 \(-0.287573\pi\)
0.618914 + 0.785459i \(0.287573\pi\)
\(138\) 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 \(-0.923015\pi\)
0.970895 0.239505i \(-0.0769852\pi\)
\(152\) 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 \(-0.870709\pi\)
0.918636 0.395104i \(-0.129291\pi\)
\(158\) 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 \(-0.501949\pi\)
−0.00612442 + 0.999981i \(0.501949\pi\)
\(180\) 0 0
\(181\) − 132.855i − 0.734003i −0.930220 0.367001i \(-0.880384\pi\)
0.930220 0.367001i \(-0.119616\pi\)
\(182\) 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 \(0.184211\pi\)
−0.837166 + 0.546949i \(0.815789\pi\)
\(200\) 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 \(-0.699776\pi\)
0.587216 0.809430i \(-0.300224\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 21.0546 0.0927515 0.0463758 0.998924i \(-0.485233\pi\)
0.0463758 + 0.998924i \(0.485233\pi\)
\(228\) 0 0
\(229\) − 319.313i − 1.39438i −0.716886 0.697191i \(-0.754433\pi\)
0.716886 0.697191i \(-0.245567\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 197.607 0.848098 0.424049 0.905639i \(-0.360608\pi\)
0.424049 + 0.905639i \(0.360608\pi\)
\(234\) 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 \(-0.180050\pi\)
0.844243 + 0.535961i \(0.180050\pi\)
\(252\) 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 \(-0.631094\pi\)
−0.400300 + 0.916384i \(0.631094\pi\)
\(258\) 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 \(0.171284\pi\)
−0.858681 + 0.512510i \(0.828716\pi\)
\(272\) 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 \(0.0536847\pi\)
−0.985811 + 0.167857i \(0.946315\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −190.784 −0.678947 −0.339474 0.940616i \(-0.610249\pi\)
−0.339474 + 0.940616i \(0.610249\pi\)
\(282\) 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 \(-0.352999\pi\)
0.445575 + 0.895244i \(0.352999\pi\)
\(348\) 0 0
\(349\) 609.516i 1.74646i 0.487306 + 0.873231i \(0.337980\pi\)
−0.487306 + 0.873231i \(0.662020\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 376.662 1.06703 0.533515 0.845791i \(-0.320871\pi\)
0.533515 + 0.845791i \(0.320871\pi\)
\(354\) 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 \(0.0387575\pi\)
−0.992596 + 0.121460i \(0.961243\pi\)
\(368\) 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 \(-0.955701\pi\)
0.990331 0.138722i \(-0.0442995\pi\)
\(374\) 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 \(-0.595527\pi\)
0.295622 0.955305i \(-0.404473\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −314.381 −0.783993 −0.391997 0.919967i \(-0.628216\pi\)
−0.391997 + 0.919967i \(0.628216\pi\)
\(402\) 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 \(-0.646277\pi\)
−0.443539 + 0.896255i \(0.646277\pi\)
\(420\) 0 0
\(421\) 660.963i 1.56998i 0.619507 + 0.784991i \(0.287333\pi\)
−0.619507 + 0.784991i \(0.712667\pi\)
\(422\) 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 \(-0.822924\pi\)
0.849214 0.528049i \(-0.177076\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −655.755 −1.48026 −0.740130 0.672464i \(-0.765236\pi\)
−0.740130 + 0.672464i \(0.765236\pi\)
\(444\) 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 \(-0.559700\pi\)
−0.186455 + 0.982464i \(0.559700\pi\)
\(450\) 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 \(0.0551958\pi\)
−0.985003 + 0.172535i \(0.944804\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −275.233 −0.589363 −0.294682 0.955595i \(-0.595214\pi\)
−0.294682 + 0.955595i \(0.595214\pi\)
\(468\) 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 \(0.0723095\pi\)
−0.974308 + 0.225218i \(0.927690\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −346.740 −0.706191 −0.353095 0.935587i \(-0.614871\pi\)
−0.353095 + 0.935587i \(0.614871\pi\)
\(492\) 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 \(-0.727820\pi\)
−0.656158 + 0.754623i \(0.727820\pi\)
\(522\) 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 \(-0.755078\pi\)
0.718296 0.695737i \(-0.244922\pi\)
\(542\) 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 \(-0.436881\pi\)
0.196997 + 0.980404i \(0.436881\pi\)
\(564\) 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 \(-0.678100\pi\)
−0.530777 + 0.847511i \(0.678100\pi\)
\(570\) 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 \(-0.599612\pi\)
−0.307858 + 0.951432i \(0.599612\pi\)
\(588\) 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 \(-0.575069\pi\)
−0.233655 + 0.972320i \(0.575069\pi\)
\(594\) 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 \(-0.844575\pi\)
0.883140 0.469109i \(-0.155425\pi\)
\(608\) 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 \(-0.625083\pi\)
0.382925 0.923779i \(-0.374917\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −651.926 −1.05661 −0.528303 0.849056i \(-0.677171\pi\)
−0.528303 + 0.849056i \(0.677171\pi\)
\(618\) 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 \(-0.705342\pi\)
0.601280 0.799038i \(-0.294658\pi\)
\(632\) 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 \(-0.657831\pi\)
−0.475770 + 0.879570i \(0.657831\pi\)
\(642\) 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 \(-0.171734\pi\)
0.857957 + 0.513722i \(0.171734\pi\)
\(660\) 0 0
\(661\) − 669.361i − 1.01265i −0.862343 0.506324i \(-0.831004\pi\)
0.862343 0.506324i \(-0.168996\pi\)
\(662\) 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 \(-0.712473\pi\)
−0.619027 + 0.785369i \(0.712473\pi\)
\(684\) 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 \(-0.799371\pi\)
0.807853 0.589383i \(-0.200629\pi\)
\(710\) 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 \(0.0608305\pi\)
−0.981795 + 0.189943i \(0.939170\pi\)
\(728\) 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 \(-0.935547\pi\)
0.979570 0.201105i \(-0.0644533\pi\)
\(734\) 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 \(-0.723211\pi\)
0.645164 0.764044i \(-0.276789\pi\)
\(752\) 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 \(0.0192377\pi\)
−0.998174 + 0.0604003i \(0.980762\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 762.676 1.00220 0.501101 0.865389i \(-0.332928\pi\)
0.501101 + 0.865389i \(0.332928\pi\)
\(762\) 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 \(-0.288142\pi\)
0.617508 + 0.786565i \(0.288142\pi\)
\(810\) 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 \(0.0361589\pi\)
−0.993555 + 0.113352i \(0.963841\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 189.836 0.229548 0.114774 0.993392i \(-0.463386\pi\)
0.114774 + 0.993392i \(0.463386\pi\)
\(828\) 0 0
\(829\) 920.478i 1.11035i 0.831735 + 0.555174i \(0.187348\pi\)
−0.831735 + 0.555174i \(0.812652\pi\)
\(830\) 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 \(0.148542\pi\)
−0.893077 + 0.449903i \(0.851458\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 30.7621 0.0358951 0.0179476 0.999839i \(-0.494287\pi\)
0.0179476 + 0.999839i \(0.494287\pi\)
\(858\) 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 \(-0.879351\pi\)
0.929024 0.370018i \(-0.120649\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 951.863 1.08043 0.540217 0.841526i \(-0.318342\pi\)
0.540217 + 0.841526i \(0.318342\pi\)
\(882\) 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 \(-0.988159\pi\)
0.999308 0.0371907i \(-0.0118409\pi\)
\(920\) 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 \(-0.445023\pi\)
0.171857 + 0.985122i \(0.445023\pi\)
\(930\) 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 \(-0.569820\pi\)
−0.217591 + 0.976040i \(0.569820\pi\)
\(948\) 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 \(-0.475057\pi\)
0.0782804 + 0.996931i \(0.475057\pi\)
\(954\) 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 \(0.0150223\pi\)
−0.998887 + 0.0471764i \(0.984978\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 117.427 0.120934 0.0604669 0.998170i \(-0.480741\pi\)
0.0604669 + 0.998170i \(0.480741\pi\)
\(972\) 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 \(-0.404718\pi\)
0.294886 + 0.955532i \(0.404718\pi\)
\(978\) 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 \(0.278160\pi\)
−0.641867 + 0.766816i \(0.721840\pi\)
\(992\) 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 \(0.0793715\pi\)
−0.969072 + 0.246777i \(0.920628\pi\)
\(998\) 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.j.1567.9 yes 12
3.2 odd 2 1728.3.b.i.1567.3 12
4.3 odd 2 inner 1728.3.b.j.1567.10 yes 12
8.3 odd 2 inner 1728.3.b.j.1567.4 yes 12
8.5 even 2 inner 1728.3.b.j.1567.3 yes 12
12.11 even 2 1728.3.b.i.1567.4 yes 12
24.5 odd 2 1728.3.b.i.1567.9 yes 12
24.11 even 2 1728.3.b.i.1567.10 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1728.3.b.i.1567.3 12 3.2 odd 2
1728.3.b.i.1567.4 yes 12 12.11 even 2
1728.3.b.i.1567.9 yes 12 24.5 odd 2
1728.3.b.i.1567.10 yes 12 24.11 even 2
1728.3.b.j.1567.3 yes 12 8.5 even 2 inner
1728.3.b.j.1567.4 yes 12 8.3 odd 2 inner
1728.3.b.j.1567.9 yes 12 1.1 even 1 trivial
1728.3.b.j.1567.10 yes 12 4.3 odd 2 inner