Properties

Label 8112.2.a.bv.1.2
Level $8112$
Weight $2$
Character 8112.1
Self dual yes
Analytic conductor $64.775$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 8112 = 2^{4} \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8112.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.7746461197\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 8112.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +3.46410 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +3.46410 q^{7} +1.00000 q^{9} +3.46410 q^{11} +6.00000 q^{17} +3.46410 q^{19} +3.46410 q^{21} -5.00000 q^{25} +1.00000 q^{27} +6.00000 q^{29} -3.46410 q^{31} +3.46410 q^{33} +6.92820 q^{37} +6.92820 q^{41} -4.00000 q^{43} -3.46410 q^{47} +5.00000 q^{49} +6.00000 q^{51} +6.00000 q^{53} +3.46410 q^{57} -10.3923 q^{59} -2.00000 q^{61} +3.46410 q^{63} -10.3923 q^{67} +3.46410 q^{71} -5.00000 q^{75} +12.0000 q^{77} +8.00000 q^{79} +1.00000 q^{81} -3.46410 q^{83} +6.00000 q^{87} -6.92820 q^{89} -3.46410 q^{93} +13.8564 q^{97} +3.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{9} + 12 q^{17} - 10 q^{25} + 2 q^{27} + 12 q^{29} - 8 q^{43} + 10 q^{49} + 12 q^{51} + 12 q^{53} - 4 q^{61} - 10 q^{75} + 24 q^{77} + 16 q^{79} + 2 q^{81} + 12 q^{87}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 3.46410 1.30931 0.654654 0.755929i \(-0.272814\pi\)
0.654654 + 0.755929i \(0.272814\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.46410 1.04447 0.522233 0.852803i \(-0.325099\pi\)
0.522233 + 0.852803i \(0.325099\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) 3.46410 0.794719 0.397360 0.917663i \(-0.369927\pi\)
0.397360 + 0.917663i \(0.369927\pi\)
\(20\) 0 0
\(21\) 3.46410 0.755929
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −3.46410 −0.622171 −0.311086 0.950382i \(-0.600693\pi\)
−0.311086 + 0.950382i \(0.600693\pi\)
\(32\) 0 0
\(33\) 3.46410 0.603023
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 6.92820 1.13899 0.569495 0.821995i \(-0.307139\pi\)
0.569495 + 0.821995i \(0.307139\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.92820 1.08200 0.541002 0.841021i \(-0.318045\pi\)
0.541002 + 0.841021i \(0.318045\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.46410 −0.505291 −0.252646 0.967559i \(-0.581301\pi\)
−0.252646 + 0.967559i \(0.581301\pi\)
\(48\) 0 0
\(49\) 5.00000 0.714286
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 3.46410 0.458831
\(58\) 0 0
\(59\) −10.3923 −1.35296 −0.676481 0.736460i \(-0.736496\pi\)
−0.676481 + 0.736460i \(0.736496\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) 3.46410 0.436436
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −10.3923 −1.26962 −0.634811 0.772667i \(-0.718922\pi\)
−0.634811 + 0.772667i \(0.718922\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3.46410 0.411113 0.205557 0.978645i \(-0.434100\pi\)
0.205557 + 0.978645i \(0.434100\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) −5.00000 −0.577350
\(76\) 0 0
\(77\) 12.0000 1.36753
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −3.46410 −0.380235 −0.190117 0.981761i \(-0.560887\pi\)
−0.190117 + 0.981761i \(0.560887\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 6.00000 0.643268
\(88\) 0 0
\(89\) −6.92820 −0.734388 −0.367194 0.930144i \(-0.619682\pi\)
−0.367194 + 0.930144i \(0.619682\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −3.46410 −0.359211
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 13.8564 1.40690 0.703452 0.710742i \(-0.251641\pi\)
0.703452 + 0.710742i \(0.251641\pi\)
\(98\) 0 0
\(99\) 3.46410 0.348155
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) −6.92820 −0.663602 −0.331801 0.943349i \(-0.607656\pi\)
−0.331801 + 0.943349i \(0.607656\pi\)
\(110\) 0 0
\(111\) 6.92820 0.657596
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 20.7846 1.90532
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 6.92820 0.624695
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 0 0
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) 12.0000 1.04053
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −20.7846 −1.77575 −0.887875 0.460086i \(-0.847819\pi\)
−0.887875 + 0.460086i \(0.847819\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) −3.46410 −0.291730
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 5.00000 0.412393
\(148\) 0 0
\(149\) −13.8564 −1.13516 −0.567581 0.823318i \(-0.692120\pi\)
−0.567581 + 0.823318i \(0.692120\pi\)
\(150\) 0 0
\(151\) −10.3923 −0.845714 −0.422857 0.906196i \(-0.638973\pi\)
−0.422857 + 0.906196i \(0.638973\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −3.46410 −0.271329 −0.135665 0.990755i \(-0.543317\pi\)
−0.135665 + 0.990755i \(0.543317\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −17.3205 −1.34030 −0.670151 0.742225i \(-0.733770\pi\)
−0.670151 + 0.742225i \(0.733770\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 3.46410 0.264906
\(172\) 0 0
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 0 0
\(175\) −17.3205 −1.30931
\(176\) 0 0
\(177\) −10.3923 −0.781133
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 0 0
\(183\) −2.00000 −0.147844
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 20.7846 1.51992
\(188\) 0 0
\(189\) 3.46410 0.251976
\(190\) 0 0
\(191\) −24.0000 −1.73658 −0.868290 0.496058i \(-0.834780\pi\)
−0.868290 + 0.496058i \(0.834780\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) −10.3923 −0.733017
\(202\) 0 0
\(203\) 20.7846 1.45879
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 12.0000 0.830057
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 0 0
\(213\) 3.46410 0.237356
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −12.0000 −0.814613
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −3.46410 −0.231973 −0.115987 0.993251i \(-0.537003\pi\)
−0.115987 + 0.993251i \(0.537003\pi\)
\(224\) 0 0
\(225\) −5.00000 −0.333333
\(226\) 0 0
\(227\) −17.3205 −1.14960 −0.574801 0.818293i \(-0.694921\pi\)
−0.574801 + 0.818293i \(0.694921\pi\)
\(228\) 0 0
\(229\) −6.92820 −0.457829 −0.228914 0.973447i \(-0.573518\pi\)
−0.228914 + 0.973447i \(0.573518\pi\)
\(230\) 0 0
\(231\) 12.0000 0.789542
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 8.00000 0.519656
\(238\) 0 0
\(239\) −10.3923 −0.672222 −0.336111 0.941822i \(-0.609112\pi\)
−0.336111 + 0.941822i \(0.609112\pi\)
\(240\) 0 0
\(241\) 13.8564 0.892570 0.446285 0.894891i \(-0.352747\pi\)
0.446285 + 0.894891i \(0.352747\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −3.46410 −0.219529
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) 24.0000 1.49129
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 0 0
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −6.92820 −0.423999
\(268\) 0 0
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) −10.3923 −0.631288 −0.315644 0.948878i \(-0.602220\pi\)
−0.315644 + 0.948878i \(0.602220\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −17.3205 −1.04447
\(276\) 0 0
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 0 0
\(279\) −3.46410 −0.207390
\(280\) 0 0
\(281\) 6.92820 0.413302 0.206651 0.978415i \(-0.433744\pi\)
0.206651 + 0.978415i \(0.433744\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 24.0000 1.41668
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 13.8564 0.812277
\(292\) 0 0
\(293\) 27.7128 1.61900 0.809500 0.587120i \(-0.199738\pi\)
0.809500 + 0.587120i \(0.199738\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 3.46410 0.201008
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −13.8564 −0.798670
\(302\) 0 0
\(303\) −6.00000 −0.344691
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 10.3923 0.593120 0.296560 0.955014i \(-0.404160\pi\)
0.296560 + 0.955014i \(0.404160\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −13.8564 −0.778253 −0.389127 0.921184i \(-0.627223\pi\)
−0.389127 + 0.921184i \(0.627223\pi\)
\(318\) 0 0
\(319\) 20.7846 1.16371
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) 20.7846 1.15649
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −6.92820 −0.383131
\(328\) 0 0
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) 3.46410 0.190404 0.0952021 0.995458i \(-0.469650\pi\)
0.0952021 + 0.995458i \(0.469650\pi\)
\(332\) 0 0
\(333\) 6.92820 0.379663
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 0 0
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) −12.0000 −0.649836
\(342\) 0 0
\(343\) −6.92820 −0.374088
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −36.0000 −1.93258 −0.966291 0.257454i \(-0.917117\pi\)
−0.966291 + 0.257454i \(0.917117\pi\)
\(348\) 0 0
\(349\) −6.92820 −0.370858 −0.185429 0.982658i \(-0.559368\pi\)
−0.185429 + 0.982658i \(0.559368\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −34.6410 −1.84376 −0.921878 0.387481i \(-0.873345\pi\)
−0.921878 + 0.387481i \(0.873345\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 20.7846 1.10004
\(358\) 0 0
\(359\) −17.3205 −0.914141 −0.457071 0.889430i \(-0.651101\pi\)
−0.457071 + 0.889430i \(0.651101\pi\)
\(360\) 0 0
\(361\) −7.00000 −0.368421
\(362\) 0 0
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 16.0000 0.835193 0.417597 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) 0 0
\(369\) 6.92820 0.360668
\(370\) 0 0
\(371\) 20.7846 1.07908
\(372\) 0 0
\(373\) 22.0000 1.13912 0.569558 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 17.3205 0.889695 0.444847 0.895606i \(-0.353258\pi\)
0.444847 + 0.895606i \(0.353258\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) 0 0
\(383\) 3.46410 0.177007 0.0885037 0.996076i \(-0.471792\pi\)
0.0885037 + 0.996076i \(0.471792\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −4.00000 −0.203331
\(388\) 0 0
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 12.0000 0.605320
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −34.6410 −1.73858 −0.869291 0.494300i \(-0.835424\pi\)
−0.869291 + 0.494300i \(0.835424\pi\)
\(398\) 0 0
\(399\) 12.0000 0.600751
\(400\) 0 0
\(401\) 6.92820 0.345978 0.172989 0.984924i \(-0.444657\pi\)
0.172989 + 0.984924i \(0.444657\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 24.0000 1.18964
\(408\) 0 0
\(409\) −27.7128 −1.37031 −0.685155 0.728397i \(-0.740266\pi\)
−0.685155 + 0.728397i \(0.740266\pi\)
\(410\) 0 0
\(411\) −20.7846 −1.02523
\(412\) 0 0
\(413\) −36.0000 −1.77144
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 4.00000 0.195881
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) 34.6410 1.68830 0.844150 0.536107i \(-0.180106\pi\)
0.844150 + 0.536107i \(0.180106\pi\)
\(422\) 0 0
\(423\) −3.46410 −0.168430
\(424\) 0 0
\(425\) −30.0000 −1.45521
\(426\) 0 0
\(427\) −6.92820 −0.335279
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −24.2487 −1.16802 −0.584010 0.811747i \(-0.698517\pi\)
−0.584010 + 0.811747i \(0.698517\pi\)
\(432\) 0 0
\(433\) −34.0000 −1.63394 −0.816968 0.576683i \(-0.804347\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) 5.00000 0.238095
\(442\) 0 0
\(443\) 36.0000 1.71041 0.855206 0.518289i \(-0.173431\pi\)
0.855206 + 0.518289i \(0.173431\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −13.8564 −0.655386
\(448\) 0 0
\(449\) 6.92820 0.326962 0.163481 0.986546i \(-0.447728\pi\)
0.163481 + 0.986546i \(0.447728\pi\)
\(450\) 0 0
\(451\) 24.0000 1.13012
\(452\) 0 0
\(453\) −10.3923 −0.488273
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 27.7128 1.29635 0.648175 0.761491i \(-0.275532\pi\)
0.648175 + 0.761491i \(0.275532\pi\)
\(458\) 0 0
\(459\) 6.00000 0.280056
\(460\) 0 0
\(461\) 13.8564 0.645357 0.322679 0.946509i \(-0.395417\pi\)
0.322679 + 0.946509i \(0.395417\pi\)
\(462\) 0 0
\(463\) 17.3205 0.804952 0.402476 0.915430i \(-0.368150\pi\)
0.402476 + 0.915430i \(0.368150\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 0 0
\(469\) −36.0000 −1.66233
\(470\) 0 0
\(471\) 14.0000 0.645086
\(472\) 0 0
\(473\) −13.8564 −0.637118
\(474\) 0 0
\(475\) −17.3205 −0.794719
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 0 0
\(479\) 10.3923 0.474837 0.237418 0.971408i \(-0.423699\pi\)
0.237418 + 0.971408i \(0.423699\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 38.1051 1.72671 0.863354 0.504599i \(-0.168360\pi\)
0.863354 + 0.504599i \(0.168360\pi\)
\(488\) 0 0
\(489\) −3.46410 −0.156652
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 0 0
\(493\) 36.0000 1.62136
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 12.0000 0.538274
\(498\) 0 0
\(499\) −10.3923 −0.465223 −0.232612 0.972570i \(-0.574727\pi\)
−0.232612 + 0.972570i \(0.574727\pi\)
\(500\) 0 0
\(501\) −17.3205 −0.773823
\(502\) 0 0
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 41.5692 1.84252 0.921262 0.388943i \(-0.127160\pi\)
0.921262 + 0.388943i \(0.127160\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 3.46410 0.152944
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −12.0000 −0.527759
\(518\) 0 0
\(519\) 18.0000 0.790112
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 0 0
\(523\) 4.00000 0.174908 0.0874539 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) 0 0
\(525\) −17.3205 −0.755929
\(526\) 0 0
\(527\) −20.7846 −0.905392
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) −10.3923 −0.450988
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −12.0000 −0.517838
\(538\) 0 0
\(539\) 17.3205 0.746047
\(540\) 0 0
\(541\) 6.92820 0.297867 0.148933 0.988847i \(-0.452416\pi\)
0.148933 + 0.988847i \(0.452416\pi\)
\(542\) 0 0
\(543\) 10.0000 0.429141
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) 0 0
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) 20.7846 0.885454
\(552\) 0 0
\(553\) 27.7128 1.17847
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 13.8564 0.587115 0.293557 0.955941i \(-0.405161\pi\)
0.293557 + 0.955941i \(0.405161\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 20.7846 0.877527
\(562\) 0 0
\(563\) −12.0000 −0.505740 −0.252870 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 3.46410 0.145479
\(568\) 0 0
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 0 0
\(573\) −24.0000 −1.00261
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −12.0000 −0.497844
\(582\) 0 0
\(583\) 20.7846 0.860811
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 10.3923 0.428936 0.214468 0.976731i \(-0.431198\pi\)
0.214468 + 0.976731i \(0.431198\pi\)
\(588\) 0 0
\(589\) −12.0000 −0.494451
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 6.92820 0.284507 0.142254 0.989830i \(-0.454565\pi\)
0.142254 + 0.989830i \(0.454565\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 16.0000 0.654836
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 0 0
\(603\) −10.3923 −0.423207
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −32.0000 −1.29884 −0.649420 0.760430i \(-0.724988\pi\)
−0.649420 + 0.760430i \(0.724988\pi\)
\(608\) 0 0
\(609\) 20.7846 0.842235
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 20.7846 0.839482 0.419741 0.907644i \(-0.362121\pi\)
0.419741 + 0.907644i \(0.362121\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 6.92820 0.278919 0.139459 0.990228i \(-0.455464\pi\)
0.139459 + 0.990228i \(0.455464\pi\)
\(618\) 0 0
\(619\) −31.1769 −1.25311 −0.626553 0.779379i \(-0.715535\pi\)
−0.626553 + 0.779379i \(0.715535\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −24.0000 −0.961540
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 12.0000 0.479234
\(628\) 0 0
\(629\) 41.5692 1.65747
\(630\) 0 0
\(631\) −38.1051 −1.51694 −0.758470 0.651707i \(-0.774053\pi\)
−0.758470 + 0.651707i \(0.774053\pi\)
\(632\) 0 0
\(633\) 20.0000 0.794929
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 3.46410 0.137038
\(640\) 0 0
\(641\) 6.00000 0.236986 0.118493 0.992955i \(-0.462194\pi\)
0.118493 + 0.992955i \(0.462194\pi\)
\(642\) 0 0
\(643\) −10.3923 −0.409832 −0.204916 0.978780i \(-0.565692\pi\)
−0.204916 + 0.978780i \(0.565692\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −24.0000 −0.943537 −0.471769 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 0 0
\(649\) −36.0000 −1.41312
\(650\) 0 0
\(651\) −12.0000 −0.470317
\(652\) 0 0
\(653\) 6.00000 0.234798 0.117399 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) −20.7846 −0.808428 −0.404214 0.914665i \(-0.632455\pi\)
−0.404214 + 0.914665i \(0.632455\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −3.46410 −0.133930
\(670\) 0 0
\(671\) −6.92820 −0.267460
\(672\) 0 0
\(673\) 46.0000 1.77317 0.886585 0.462566i \(-0.153071\pi\)
0.886585 + 0.462566i \(0.153071\pi\)
\(674\) 0 0
\(675\) −5.00000 −0.192450
\(676\) 0 0
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 0 0
\(679\) 48.0000 1.84207
\(680\) 0 0
\(681\) −17.3205 −0.663723
\(682\) 0 0
\(683\) 31.1769 1.19295 0.596476 0.802631i \(-0.296567\pi\)
0.596476 + 0.802631i \(0.296567\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −6.92820 −0.264327
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 45.0333 1.71315 0.856574 0.516024i \(-0.172588\pi\)
0.856574 + 0.516024i \(0.172588\pi\)
\(692\) 0 0
\(693\) 12.0000 0.455842
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 41.5692 1.57455
\(698\) 0 0
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) 42.0000 1.58632 0.793159 0.609015i \(-0.208435\pi\)
0.793159 + 0.609015i \(0.208435\pi\)
\(702\) 0 0
\(703\) 24.0000 0.905177
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −20.7846 −0.781686
\(708\) 0 0
\(709\) −6.92820 −0.260194 −0.130097 0.991501i \(-0.541529\pi\)
−0.130097 + 0.991501i \(0.541529\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −10.3923 −0.388108
\(718\) 0 0
\(719\) 24.0000 0.895049 0.447524 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 0 0
\(721\) 27.7128 1.03208
\(722\) 0 0
\(723\) 13.8564 0.515325
\(724\) 0 0
\(725\) −30.0000 −1.11417
\(726\) 0 0
\(727\) 16.0000 0.593407 0.296704 0.954970i \(-0.404113\pi\)
0.296704 + 0.954970i \(0.404113\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −24.0000 −0.887672
\(732\) 0 0
\(733\) −34.6410 −1.27950 −0.639748 0.768585i \(-0.720961\pi\)
−0.639748 + 0.768585i \(0.720961\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −36.0000 −1.32608
\(738\) 0 0
\(739\) 38.1051 1.40172 0.700860 0.713299i \(-0.252800\pi\)
0.700860 + 0.713299i \(0.252800\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 3.46410 0.127086 0.0635428 0.997979i \(-0.479760\pi\)
0.0635428 + 0.997979i \(0.479760\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −3.46410 −0.126745
\(748\) 0 0
\(749\) −41.5692 −1.51891
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 0 0
\(753\) −12.0000 −0.437304
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 22.0000 0.799604 0.399802 0.916602i \(-0.369079\pi\)
0.399802 + 0.916602i \(0.369079\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −48.4974 −1.75803 −0.879015 0.476794i \(-0.841799\pi\)
−0.879015 + 0.476794i \(0.841799\pi\)
\(762\) 0 0
\(763\) −24.0000 −0.868858
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −27.7128 −0.999350 −0.499675 0.866213i \(-0.666547\pi\)
−0.499675 + 0.866213i \(0.666547\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) 0 0
\(773\) 13.8564 0.498380 0.249190 0.968455i \(-0.419836\pi\)
0.249190 + 0.968455i \(0.419836\pi\)
\(774\) 0 0
\(775\) 17.3205 0.622171
\(776\) 0 0
\(777\) 24.0000 0.860995
\(778\) 0 0
\(779\) 24.0000 0.859889
\(780\) 0 0
\(781\) 12.0000 0.429394
\(782\) 0 0
\(783\) 6.00000 0.214423
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 10.3923 0.370446 0.185223 0.982697i \(-0.440699\pi\)
0.185223 + 0.982697i \(0.440699\pi\)
\(788\) 0 0
\(789\) 24.0000 0.854423
\(790\) 0 0
\(791\) −20.7846 −0.739016
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 42.0000 1.48772 0.743858 0.668338i \(-0.232994\pi\)
0.743858 + 0.668338i \(0.232994\pi\)
\(798\) 0 0
\(799\) −20.7846 −0.735307
\(800\) 0 0
\(801\) −6.92820 −0.244796
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 6.00000 0.211210
\(808\) 0 0
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 0 0
\(811\) −38.1051 −1.33805 −0.669026 0.743239i \(-0.733288\pi\)
−0.669026 + 0.743239i \(0.733288\pi\)
\(812\) 0 0
\(813\) −10.3923 −0.364474
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −13.8564 −0.484774
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −13.8564 −0.483592 −0.241796 0.970327i \(-0.577736\pi\)
−0.241796 + 0.970327i \(0.577736\pi\)
\(822\) 0 0
\(823\) −40.0000 −1.39431 −0.697156 0.716919i \(-0.745552\pi\)
−0.697156 + 0.716919i \(0.745552\pi\)
\(824\) 0 0
\(825\) −17.3205 −0.603023
\(826\) 0 0
\(827\) −24.2487 −0.843210 −0.421605 0.906780i \(-0.638533\pi\)
−0.421605 + 0.906780i \(0.638533\pi\)
\(828\) 0 0
\(829\) 2.00000 0.0694629 0.0347314 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) 0 0
\(831\) 10.0000 0.346896
\(832\) 0 0
\(833\) 30.0000 1.03944
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −3.46410 −0.119737
\(838\) 0 0
\(839\) −3.46410 −0.119594 −0.0597970 0.998211i \(-0.519045\pi\)
−0.0597970 + 0.998211i \(0.519045\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 6.92820 0.238620
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 3.46410 0.119028
\(848\) 0 0
\(849\) 4.00000 0.137280
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −20.7846 −0.711651 −0.355826 0.934552i \(-0.615800\pi\)
−0.355826 + 0.934552i \(0.615800\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −42.0000 −1.43469 −0.717346 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) 0 0
\(859\) −4.00000 −0.136478 −0.0682391 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) 0 0
\(861\) 24.0000 0.817918
\(862\) 0 0
\(863\) 31.1769 1.06127 0.530637 0.847599i \(-0.321953\pi\)
0.530637 + 0.847599i \(0.321953\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 19.0000 0.645274
\(868\) 0 0
\(869\) 27.7128 0.940093
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 13.8564 0.468968
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −48.4974 −1.63764 −0.818821 0.574049i \(-0.805372\pi\)
−0.818821 + 0.574049i \(0.805372\pi\)
\(878\) 0 0
\(879\) 27.7128 0.934730
\(880\) 0 0
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 0 0
\(883\) −20.0000 −0.673054 −0.336527 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) 0 0
\(889\) −27.7128 −0.929458
\(890\) 0 0
\(891\) 3.46410 0.116052
\(892\) 0 0
\(893\) −12.0000 −0.401565
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −20.7846 −0.693206
\(900\) 0 0
\(901\) 36.0000 1.19933
\(902\) 0 0
\(903\) −13.8564 −0.461112
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −44.0000 −1.46100 −0.730498 0.682915i \(-0.760712\pi\)
−0.730498 + 0.682915i \(0.760712\pi\)
\(908\) 0 0
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) −24.0000 −0.795155 −0.397578 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) 0 0
\(913\) −12.0000 −0.397142
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 41.5692 1.37274
\(918\) 0 0
\(919\) 32.0000 1.05558 0.527791 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 0 0
\(921\) 10.3923 0.342438
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −34.6410 −1.13899
\(926\) 0 0
\(927\) 8.00000 0.262754
\(928\) 0 0
\(929\) 20.7846 0.681921 0.340960 0.940078i \(-0.389248\pi\)
0.340960 + 0.940078i \(0.389248\pi\)
\(930\) 0 0
\(931\) 17.3205 0.567657
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −38.0000 −1.24141 −0.620703 0.784046i \(-0.713153\pi\)
−0.620703 + 0.784046i \(0.713153\pi\)
\(938\) 0 0
\(939\) 10.0000 0.326338
\(940\) 0 0
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −51.9615 −1.68852 −0.844261 0.535932i \(-0.819960\pi\)
−0.844261 + 0.535932i \(0.819960\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −13.8564 −0.449325
\(952\) 0 0
\(953\) −42.0000 −1.36051 −0.680257 0.732974i \(-0.738132\pi\)
−0.680257 + 0.732974i \(0.738132\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 20.7846 0.671871
\(958\) 0 0
\(959\) −72.0000 −2.32500
\(960\) 0 0
\(961\) −19.0000 −0.612903
\(962\) 0 0
\(963\) −12.0000 −0.386695
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 10.3923 0.334194 0.167097 0.985940i \(-0.446561\pi\)
0.167097 + 0.985940i \(0.446561\pi\)
\(968\) 0 0
\(969\) 20.7846 0.667698
\(970\) 0 0
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 0 0
\(973\) 13.8564 0.444216
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −48.4974 −1.55157 −0.775785 0.630997i \(-0.782646\pi\)
−0.775785 + 0.630997i \(0.782646\pi\)
\(978\) 0 0
\(979\) −24.0000 −0.767043
\(980\) 0 0
\(981\) −6.92820 −0.221201
\(982\) 0 0
\(983\) 51.9615 1.65732 0.828658 0.559756i \(-0.189105\pi\)
0.828658 + 0.559756i \(0.189105\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −12.0000 −0.381964
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −32.0000 −1.01651 −0.508257 0.861206i \(-0.669710\pi\)
−0.508257 + 0.861206i \(0.669710\pi\)
\(992\) 0 0
\(993\) 3.46410 0.109930
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 38.0000 1.20347 0.601736 0.798695i \(-0.294476\pi\)
0.601736 + 0.798695i \(0.294476\pi\)
\(998\) 0 0
\(999\) 6.92820 0.219199
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 8112.2.a.bv.1.2 2
4.3 odd 2 507.2.a.f.1.1 2
12.11 even 2 1521.2.a.l.1.2 2
13.5 odd 4 624.2.c.e.337.2 2
13.8 odd 4 624.2.c.e.337.1 2
13.12 even 2 inner 8112.2.a.bv.1.1 2
39.5 even 4 1872.2.c.e.1585.2 2
39.8 even 4 1872.2.c.e.1585.1 2
52.3 odd 6 507.2.e.e.22.2 4
52.7 even 12 507.2.j.a.361.1 2
52.11 even 12 507.2.j.c.316.1 2
52.15 even 12 507.2.j.a.316.1 2
52.19 even 12 507.2.j.c.361.1 2
52.23 odd 6 507.2.e.e.22.1 4
52.31 even 4 39.2.b.a.25.2 yes 2
52.35 odd 6 507.2.e.e.484.2 4
52.43 odd 6 507.2.e.e.484.1 4
52.47 even 4 39.2.b.a.25.1 2
52.51 odd 2 507.2.a.f.1.2 2
104.5 odd 4 2496.2.c.d.961.2 2
104.21 odd 4 2496.2.c.d.961.1 2
104.83 even 4 2496.2.c.k.961.1 2
104.99 even 4 2496.2.c.k.961.2 2
156.47 odd 4 117.2.b.a.64.2 2
156.83 odd 4 117.2.b.a.64.1 2
156.155 even 2 1521.2.a.l.1.1 2
260.47 odd 4 975.2.h.f.649.4 4
260.83 odd 4 975.2.h.f.649.3 4
260.99 even 4 975.2.b.d.376.2 2
260.187 odd 4 975.2.h.f.649.2 4
260.203 odd 4 975.2.h.f.649.1 4
260.239 even 4 975.2.b.d.376.1 2
364.83 odd 4 1911.2.c.d.883.2 2
364.307 odd 4 1911.2.c.d.883.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.2.b.a.25.1 2 52.47 even 4
39.2.b.a.25.2 yes 2 52.31 even 4
117.2.b.a.64.1 2 156.83 odd 4
117.2.b.a.64.2 2 156.47 odd 4
507.2.a.f.1.1 2 4.3 odd 2
507.2.a.f.1.2 2 52.51 odd 2
507.2.e.e.22.1 4 52.23 odd 6
507.2.e.e.22.2 4 52.3 odd 6
507.2.e.e.484.1 4 52.43 odd 6
507.2.e.e.484.2 4 52.35 odd 6
507.2.j.a.316.1 2 52.15 even 12
507.2.j.a.361.1 2 52.7 even 12
507.2.j.c.316.1 2 52.11 even 12
507.2.j.c.361.1 2 52.19 even 12
624.2.c.e.337.1 2 13.8 odd 4
624.2.c.e.337.2 2 13.5 odd 4
975.2.b.d.376.1 2 260.239 even 4
975.2.b.d.376.2 2 260.99 even 4
975.2.h.f.649.1 4 260.203 odd 4
975.2.h.f.649.2 4 260.187 odd 4
975.2.h.f.649.3 4 260.83 odd 4
975.2.h.f.649.4 4 260.47 odd 4
1521.2.a.l.1.1 2 156.155 even 2
1521.2.a.l.1.2 2 12.11 even 2
1872.2.c.e.1585.1 2 39.8 even 4
1872.2.c.e.1585.2 2 39.5 even 4
1911.2.c.d.883.1 2 364.307 odd 4
1911.2.c.d.883.2 2 364.83 odd 4
2496.2.c.d.961.1 2 104.21 odd 4
2496.2.c.d.961.2 2 104.5 odd 4
2496.2.c.k.961.1 2 104.83 even 4
2496.2.c.k.961.2 2 104.99 even 4
8112.2.a.bv.1.1 2 13.12 even 2 inner
8112.2.a.bv.1.2 2 1.1 even 1 trivial