Properties

Label 4056.2.c.d.337.1
Level $4056$
Weight $2$
Character 4056.337
Analytic conductor $32.387$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4056.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.3873230598\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 312)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 4056.337
Dual form 4056.2.c.d.337.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -4.00000i q^{5} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -4.00000i q^{5} +1.00000 q^{9} -2.00000i q^{11} +4.00000i q^{15} -2.00000 q^{17} -8.00000i q^{19} -4.00000 q^{23} -11.0000 q^{25} -1.00000 q^{27} -6.00000 q^{29} +4.00000i q^{31} +2.00000i q^{33} +6.00000i q^{37} +12.0000i q^{41} -4.00000 q^{43} -4.00000i q^{45} -6.00000i q^{47} +7.00000 q^{49} +2.00000 q^{51} -2.00000 q^{53} -8.00000 q^{55} +8.00000i q^{57} -14.0000i q^{59} +10.0000 q^{61} +4.00000i q^{67} +4.00000 q^{69} -2.00000i q^{71} -2.00000i q^{73} +11.0000 q^{75} -8.00000 q^{79} +1.00000 q^{81} -14.0000i q^{83} +8.00000i q^{85} +6.00000 q^{87} -4.00000i q^{93} -32.0000 q^{95} +10.0000i q^{97} -2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{9} - 4 q^{17} - 8 q^{23} - 22 q^{25} - 2 q^{27} - 12 q^{29} - 8 q^{43} + 14 q^{49} + 4 q^{51} - 4 q^{53} - 16 q^{55} + 20 q^{61} + 8 q^{69} + 22 q^{75} - 16 q^{79} + 2 q^{81} + 12 q^{87} - 64 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4056\mathbb{Z}\right)^\times\).

\(n\) \(1015\) \(2029\) \(2705\) \(3889\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) − 4.00000i − 1.78885i −0.447214 0.894427i \(-0.647584\pi\)
0.447214 0.894427i \(-0.352416\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) − 2.00000i − 0.603023i −0.953463 0.301511i \(-0.902509\pi\)
0.953463 0.301511i \(-0.0974911\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) 4.00000i 1.03280i
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) − 8.00000i − 1.83533i −0.397360 0.917663i \(-0.630073\pi\)
0.397360 0.917663i \(-0.369927\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) −11.0000 −2.20000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 4.00000i 0.718421i 0.933257 + 0.359211i \(0.116954\pi\)
−0.933257 + 0.359211i \(0.883046\pi\)
\(32\) 0 0
\(33\) 2.00000i 0.348155i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 6.00000i 0.986394i 0.869918 + 0.493197i \(0.164172\pi\)
−0.869918 + 0.493197i \(0.835828\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 12.0000i 1.87409i 0.349215 + 0.937043i \(0.386448\pi\)
−0.349215 + 0.937043i \(0.613552\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) − 4.00000i − 0.596285i
\(46\) 0 0
\(47\) − 6.00000i − 0.875190i −0.899172 0.437595i \(-0.855830\pi\)
0.899172 0.437595i \(-0.144170\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) −8.00000 −1.07872
\(56\) 0 0
\(57\) 8.00000i 1.05963i
\(58\) 0 0
\(59\) − 14.0000i − 1.82264i −0.411693 0.911322i \(-0.635063\pi\)
0.411693 0.911322i \(-0.364937\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) 0 0
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) − 2.00000i − 0.237356i −0.992933 0.118678i \(-0.962134\pi\)
0.992933 0.118678i \(-0.0378657\pi\)
\(72\) 0 0
\(73\) − 2.00000i − 0.234082i −0.993127 0.117041i \(-0.962659\pi\)
0.993127 0.117041i \(-0.0373409\pi\)
\(74\) 0 0
\(75\) 11.0000 1.27017
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 14.0000i − 1.53670i −0.640030 0.768350i \(-0.721078\pi\)
0.640030 0.768350i \(-0.278922\pi\)
\(84\) 0 0
\(85\) 8.00000i 0.867722i
\(86\) 0 0
\(87\) 6.00000 0.643268
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) − 4.00000i − 0.414781i
\(94\) 0 0
\(95\) −32.0000 −3.28313
\(96\) 0 0
\(97\) 10.0000i 1.01535i 0.861550 + 0.507673i \(0.169494\pi\)
−0.861550 + 0.507673i \(0.830506\pi\)
\(98\) 0 0
\(99\) − 2.00000i − 0.201008i
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 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
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) − 10.0000i − 0.957826i −0.877862 0.478913i \(-0.841031\pi\)
0.877862 0.478913i \(-0.158969\pi\)
\(110\) 0 0
\(111\) − 6.00000i − 0.569495i
\(112\) 0 0
\(113\) −18.0000 −1.69330 −0.846649 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) 0 0
\(115\) 16.0000i 1.49201i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 7.00000 0.636364
\(122\) 0 0
\(123\) − 12.0000i − 1.08200i
\(124\) 0 0
\(125\) 24.0000i 2.14663i
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 0 0
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 8.00000 0.698963 0.349482 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 4.00000i 0.344265i
\(136\) 0 0
\(137\) − 12.0000i − 1.02523i −0.858619 0.512615i \(-0.828677\pi\)
0.858619 0.512615i \(-0.171323\pi\)
\(138\) 0 0
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) 6.00000i 0.505291i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 24.0000i 1.99309i
\(146\) 0 0
\(147\) −7.00000 −0.577350
\(148\) 0 0
\(149\) 16.0000i 1.31077i 0.755295 + 0.655386i \(0.227494\pi\)
−0.755295 + 0.655386i \(0.772506\pi\)
\(150\) 0 0
\(151\) 20.0000i 1.62758i 0.581161 + 0.813788i \(0.302599\pi\)
−0.581161 + 0.813788i \(0.697401\pi\)
\(152\) 0 0
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) 16.0000 1.28515
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 0 0
\(159\) 2.00000 0.158610
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 24.0000i 1.87983i 0.341415 + 0.939913i \(0.389094\pi\)
−0.341415 + 0.939913i \(0.610906\pi\)
\(164\) 0 0
\(165\) 8.00000 0.622799
\(166\) 0 0
\(167\) 18.0000i 1.39288i 0.717614 + 0.696441i \(0.245234\pi\)
−0.717614 + 0.696441i \(0.754766\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) − 8.00000i − 0.611775i
\(172\) 0 0
\(173\) −18.0000 −1.36851 −0.684257 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 14.0000i 1.05230i
\(178\) 0 0
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 0 0
\(183\) −10.0000 −0.739221
\(184\) 0 0
\(185\) 24.0000 1.76452
\(186\) 0 0
\(187\) 4.00000i 0.292509i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 26.0000i 1.87152i 0.352636 + 0.935760i \(0.385285\pi\)
−0.352636 + 0.935760i \(0.614715\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 12.0000i − 0.854965i −0.904024 0.427482i \(-0.859401\pi\)
0.904024 0.427482i \(-0.140599\pi\)
\(198\) 0 0
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 0 0
\(201\) − 4.00000i − 0.282138i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 48.0000 3.35247
\(206\) 0 0
\(207\) −4.00000 −0.278019
\(208\) 0 0
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 0 0
\(213\) 2.00000i 0.137038i
\(214\) 0 0
\(215\) 16.0000i 1.09119i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 2.00000i 0.135147i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) − 28.0000i − 1.87502i −0.347960 0.937509i \(-0.613126\pi\)
0.347960 0.937509i \(-0.386874\pi\)
\(224\) 0 0
\(225\) −11.0000 −0.733333
\(226\) 0 0
\(227\) 6.00000i 0.398234i 0.979976 + 0.199117i \(0.0638074\pi\)
−0.979976 + 0.199117i \(0.936193\pi\)
\(228\) 0 0
\(229\) 10.0000i 0.660819i 0.943838 + 0.330409i \(0.107187\pi\)
−0.943838 + 0.330409i \(0.892813\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 2.00000 0.131024 0.0655122 0.997852i \(-0.479132\pi\)
0.0655122 + 0.997852i \(0.479132\pi\)
\(234\) 0 0
\(235\) −24.0000 −1.56559
\(236\) 0 0
\(237\) 8.00000 0.519656
\(238\) 0 0
\(239\) − 2.00000i − 0.129369i −0.997906 0.0646846i \(-0.979396\pi\)
0.997906 0.0646846i \(-0.0206041\pi\)
\(240\) 0 0
\(241\) − 18.0000i − 1.15948i −0.814801 0.579741i \(-0.803154\pi\)
0.814801 0.579741i \(-0.196846\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) − 28.0000i − 1.78885i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 14.0000i 0.887214i
\(250\) 0 0
\(251\) −8.00000 −0.504956 −0.252478 0.967603i \(-0.581245\pi\)
−0.252478 + 0.967603i \(0.581245\pi\)
\(252\) 0 0
\(253\) 8.00000i 0.502956i
\(254\) 0 0
\(255\) − 8.00000i − 0.500979i
\(256\) 0 0
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) −20.0000 −1.23325 −0.616626 0.787256i \(-0.711501\pi\)
−0.616626 + 0.787256i \(0.711501\pi\)
\(264\) 0 0
\(265\) 8.00000i 0.491436i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 26.0000 1.58525 0.792624 0.609711i \(-0.208714\pi\)
0.792624 + 0.609711i \(0.208714\pi\)
\(270\) 0 0
\(271\) 8.00000i 0.485965i 0.970031 + 0.242983i \(0.0781258\pi\)
−0.970031 + 0.242983i \(0.921874\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 22.0000i 1.32665i
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 0 0
\(279\) 4.00000i 0.239474i
\(280\) 0 0
\(281\) 20.0000i 1.19310i 0.802576 + 0.596550i \(0.203462\pi\)
−0.802576 + 0.596550i \(0.796538\pi\)
\(282\) 0 0
\(283\) −20.0000 −1.18888 −0.594438 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) 0 0
\(285\) 32.0000 1.89552
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) − 10.0000i − 0.586210i
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) −56.0000 −3.26045
\(296\) 0 0
\(297\) 2.00000i 0.116052i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −6.00000 −0.344691
\(304\) 0 0
\(305\) − 40.0000i − 2.29039i
\(306\) 0 0
\(307\) − 12.0000i − 0.684876i −0.939540 0.342438i \(-0.888747\pi\)
0.939540 0.342438i \(-0.111253\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 0 0
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 12.0000i 0.673987i 0.941507 + 0.336994i \(0.109410\pi\)
−0.941507 + 0.336994i \(0.890590\pi\)
\(318\) 0 0
\(319\) 12.0000i 0.671871i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 16.0000i 0.890264i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 10.0000i 0.553001i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 28.0000i 1.53902i 0.638635 + 0.769510i \(0.279499\pi\)
−0.638635 + 0.769510i \(0.720501\pi\)
\(332\) 0 0
\(333\) 6.00000i 0.328798i
\(334\) 0 0
\(335\) 16.0000 0.874173
\(336\) 0 0
\(337\) −30.0000 −1.63420 −0.817102 0.576493i \(-0.804421\pi\)
−0.817102 + 0.576493i \(0.804421\pi\)
\(338\) 0 0
\(339\) 18.0000 0.977626
\(340\) 0 0
\(341\) 8.00000 0.433224
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) − 16.0000i − 0.861411i
\(346\) 0 0
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 0 0
\(349\) 6.00000i 0.321173i 0.987022 + 0.160586i \(0.0513385\pi\)
−0.987022 + 0.160586i \(0.948662\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 28.0000i 1.49029i 0.666903 + 0.745145i \(0.267620\pi\)
−0.666903 + 0.745145i \(0.732380\pi\)
\(354\) 0 0
\(355\) −8.00000 −0.424596
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 30.0000i − 1.58334i −0.610949 0.791670i \(-0.709212\pi\)
0.610949 0.791670i \(-0.290788\pi\)
\(360\) 0 0
\(361\) −45.0000 −2.36842
\(362\) 0 0
\(363\) −7.00000 −0.367405
\(364\) 0 0
\(365\) −8.00000 −0.418739
\(366\) 0 0
\(367\) −8.00000 −0.417597 −0.208798 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 0 0
\(369\) 12.0000i 0.624695i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 26.0000 1.34623 0.673114 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) 0 0
\(375\) − 24.0000i − 1.23935i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 16.0000i 0.821865i 0.911666 + 0.410932i \(0.134797\pi\)
−0.911666 + 0.410932i \(0.865203\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) 0 0
\(383\) − 30.0000i − 1.53293i −0.642287 0.766464i \(-0.722014\pi\)
0.642287 0.766464i \(-0.277986\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −4.00000 −0.203331
\(388\) 0 0
\(389\) 26.0000 1.31825 0.659126 0.752032i \(-0.270926\pi\)
0.659126 + 0.752032i \(0.270926\pi\)
\(390\) 0 0
\(391\) 8.00000 0.404577
\(392\) 0 0
\(393\) −8.00000 −0.403547
\(394\) 0 0
\(395\) 32.0000i 1.61009i
\(396\) 0 0
\(397\) − 10.0000i − 0.501886i −0.968002 0.250943i \(-0.919259\pi\)
0.968002 0.250943i \(-0.0807406\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) − 4.00000i − 0.198762i
\(406\) 0 0
\(407\) 12.0000 0.594818
\(408\) 0 0
\(409\) − 6.00000i − 0.296681i −0.988936 0.148340i \(-0.952607\pi\)
0.988936 0.148340i \(-0.0473931\pi\)
\(410\) 0 0
\(411\) 12.0000i 0.591916i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −56.0000 −2.74893
\(416\) 0 0
\(417\) 12.0000 0.587643
\(418\) 0 0
\(419\) −24.0000 −1.17248 −0.586238 0.810139i \(-0.699392\pi\)
−0.586238 + 0.810139i \(0.699392\pi\)
\(420\) 0 0
\(421\) − 26.0000i − 1.26716i −0.773676 0.633581i \(-0.781584\pi\)
0.773676 0.633581i \(-0.218416\pi\)
\(422\) 0 0
\(423\) − 6.00000i − 0.291730i
\(424\) 0 0
\(425\) 22.0000 1.06716
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 14.0000i − 0.674356i −0.941441 0.337178i \(-0.890528\pi\)
0.941441 0.337178i \(-0.109472\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) − 24.0000i − 1.15071i
\(436\) 0 0
\(437\) 32.0000i 1.53077i
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 7.00000 0.333333
\(442\) 0 0
\(443\) 8.00000 0.380091 0.190046 0.981775i \(-0.439136\pi\)
0.190046 + 0.981775i \(0.439136\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 16.0000i − 0.756774i
\(448\) 0 0
\(449\) − 8.00000i − 0.377543i −0.982021 0.188772i \(-0.939549\pi\)
0.982021 0.188772i \(-0.0604506\pi\)
\(450\) 0 0
\(451\) 24.0000 1.13012
\(452\) 0 0
\(453\) − 20.0000i − 0.939682i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 10.0000i 0.467780i 0.972263 + 0.233890i \(0.0751456\pi\)
−0.972263 + 0.233890i \(0.924854\pi\)
\(458\) 0 0
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) − 28.0000i − 1.30409i −0.758180 0.652045i \(-0.773911\pi\)
0.758180 0.652045i \(-0.226089\pi\)
\(462\) 0 0
\(463\) − 16.0000i − 0.743583i −0.928316 0.371792i \(-0.878744\pi\)
0.928316 0.371792i \(-0.121256\pi\)
\(464\) 0 0
\(465\) −16.0000 −0.741982
\(466\) 0 0
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 2.00000 0.0921551
\(472\) 0 0
\(473\) 8.00000i 0.367840i
\(474\) 0 0
\(475\) 88.0000i 4.03772i
\(476\) 0 0
\(477\) −2.00000 −0.0915737
\(478\) 0 0
\(479\) 14.0000i 0.639676i 0.947472 + 0.319838i \(0.103629\pi\)
−0.947472 + 0.319838i \(0.896371\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 40.0000 1.81631
\(486\) 0 0
\(487\) − 28.0000i − 1.26880i −0.773004 0.634401i \(-0.781247\pi\)
0.773004 0.634401i \(-0.218753\pi\)
\(488\) 0 0
\(489\) − 24.0000i − 1.08532i
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 12.0000 0.540453
\(494\) 0 0
\(495\) −8.00000 −0.359573
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) − 36.0000i − 1.61158i −0.592200 0.805791i \(-0.701741\pi\)
0.592200 0.805791i \(-0.298259\pi\)
\(500\) 0 0
\(501\) − 18.0000i − 0.804181i
\(502\) 0 0
\(503\) 36.0000 1.60516 0.802580 0.596544i \(-0.203460\pi\)
0.802580 + 0.596544i \(0.203460\pi\)
\(504\) 0 0
\(505\) − 24.0000i − 1.06799i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 12.0000i − 0.531891i −0.963988 0.265945i \(-0.914316\pi\)
0.963988 0.265945i \(-0.0856841\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 8.00000i 0.353209i
\(514\) 0 0
\(515\) − 32.0000i − 1.41009i
\(516\) 0 0
\(517\) −12.0000 −0.527759
\(518\) 0 0
\(519\) 18.0000 0.790112
\(520\) 0 0
\(521\) −10.0000 −0.438108 −0.219054 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) 0 0
\(523\) −20.0000 −0.874539 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 8.00000i − 0.348485i
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) − 14.0000i − 0.607548i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −12.0000 −0.517838
\(538\) 0 0
\(539\) − 14.0000i − 0.603023i
\(540\) 0 0
\(541\) − 6.00000i − 0.257960i −0.991647 0.128980i \(-0.958830\pi\)
0.991647 0.128980i \(-0.0411703\pi\)
\(542\) 0 0
\(543\) 6.00000 0.257485
\(544\) 0 0
\(545\) −40.0000 −1.71341
\(546\) 0 0
\(547\) −4.00000 −0.171028 −0.0855138 0.996337i \(-0.527253\pi\)
−0.0855138 + 0.996337i \(0.527253\pi\)
\(548\) 0 0
\(549\) 10.0000 0.426790
\(550\) 0 0
\(551\) 48.0000i 2.04487i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −24.0000 −1.01874
\(556\) 0 0
\(557\) − 8.00000i − 0.338971i −0.985533 0.169485i \(-0.945789\pi\)
0.985533 0.169485i \(-0.0542106\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) − 4.00000i − 0.168880i
\(562\) 0 0
\(563\) −4.00000 −0.168580 −0.0842900 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) 0 0
\(565\) 72.0000i 3.02906i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 2.00000 0.0838444 0.0419222 0.999121i \(-0.486652\pi\)
0.0419222 + 0.999121i \(0.486652\pi\)
\(570\) 0 0
\(571\) −12.0000 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 44.0000 1.83493
\(576\) 0 0
\(577\) − 2.00000i − 0.0832611i −0.999133 0.0416305i \(-0.986745\pi\)
0.999133 0.0416305i \(-0.0132552\pi\)
\(578\) 0 0
\(579\) − 26.0000i − 1.08052i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 4.00000i 0.165663i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 22.0000i − 0.908037i −0.890992 0.454019i \(-0.849990\pi\)
0.890992 0.454019i \(-0.150010\pi\)
\(588\) 0 0
\(589\) 32.0000 1.31854
\(590\) 0 0
\(591\) 12.0000i 0.493614i
\(592\) 0 0
\(593\) − 8.00000i − 0.328521i −0.986417 0.164260i \(-0.947476\pi\)
0.986417 0.164260i \(-0.0525237\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −8.00000 −0.327418
\(598\) 0 0
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) 0 0
\(601\) 22.0000 0.897399 0.448699 0.893683i \(-0.351887\pi\)
0.448699 + 0.893683i \(0.351887\pi\)
\(602\) 0 0
\(603\) 4.00000i 0.162893i
\(604\) 0 0
\(605\) − 28.0000i − 1.13836i
\(606\) 0 0
\(607\) −40.0000 −1.62355 −0.811775 0.583970i \(-0.801498\pi\)
−0.811775 + 0.583970i \(0.801498\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) − 14.0000i − 0.565455i −0.959200 0.282727i \(-0.908761\pi\)
0.959200 0.282727i \(-0.0912392\pi\)
\(614\) 0 0
\(615\) −48.0000 −1.93555
\(616\) 0 0
\(617\) − 12.0000i − 0.483102i −0.970388 0.241551i \(-0.922344\pi\)
0.970388 0.241551i \(-0.0776561\pi\)
\(618\) 0 0
\(619\) 28.0000i 1.12542i 0.826656 + 0.562708i \(0.190240\pi\)
−0.826656 + 0.562708i \(0.809760\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 41.0000 1.64000
\(626\) 0 0
\(627\) 16.0000 0.638978
\(628\) 0 0
\(629\) − 12.0000i − 0.478471i
\(630\) 0 0
\(631\) − 20.0000i − 0.796187i −0.917345 0.398094i \(-0.869672\pi\)
0.917345 0.398094i \(-0.130328\pi\)
\(632\) 0 0
\(633\) 20.0000 0.794929
\(634\) 0 0
\(635\) − 32.0000i − 1.26988i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) − 2.00000i − 0.0791188i
\(640\) 0 0
\(641\) 14.0000 0.552967 0.276483 0.961019i \(-0.410831\pi\)
0.276483 + 0.961019i \(0.410831\pi\)
\(642\) 0 0
\(643\) 36.0000i 1.41970i 0.704352 + 0.709851i \(0.251238\pi\)
−0.704352 + 0.709851i \(0.748762\pi\)
\(644\) 0 0
\(645\) − 16.0000i − 0.629999i
\(646\) 0 0
\(647\) −32.0000 −1.25805 −0.629025 0.777385i \(-0.716546\pi\)
−0.629025 + 0.777385i \(0.716546\pi\)
\(648\) 0 0
\(649\) −28.0000 −1.09910
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −6.00000 −0.234798 −0.117399 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 0 0
\(655\) − 32.0000i − 1.25034i
\(656\) 0 0
\(657\) − 2.00000i − 0.0780274i
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) 38.0000i 1.47803i 0.673690 + 0.739014i \(0.264708\pi\)
−0.673690 + 0.739014i \(0.735292\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 24.0000 0.929284
\(668\) 0 0
\(669\) 28.0000i 1.08254i
\(670\) 0 0
\(671\) − 20.0000i − 0.772091i
\(672\) 0 0
\(673\) 14.0000 0.539660 0.269830 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) 0 0
\(675\) 11.0000 0.423390
\(676\) 0 0
\(677\) −26.0000 −0.999261 −0.499631 0.866239i \(-0.666531\pi\)
−0.499631 + 0.866239i \(0.666531\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) − 6.00000i − 0.229920i
\(682\) 0 0
\(683\) − 2.00000i − 0.0765279i −0.999268 0.0382639i \(-0.987817\pi\)
0.999268 0.0382639i \(-0.0121828\pi\)
\(684\) 0 0
\(685\) −48.0000 −1.83399
\(686\) 0 0
\(687\) − 10.0000i − 0.381524i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) − 20.0000i − 0.760836i −0.924815 0.380418i \(-0.875780\pi\)
0.924815 0.380418i \(-0.124220\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 48.0000i 1.82074i
\(696\) 0 0
\(697\) − 24.0000i − 0.909065i
\(698\) 0 0
\(699\) −2.00000 −0.0756469
\(700\) 0 0
\(701\) −6.00000 −0.226617 −0.113308 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) 0 0
\(703\) 48.0000 1.81035
\(704\) 0 0
\(705\) 24.0000 0.903892
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 2.00000i 0.0751116i 0.999295 + 0.0375558i \(0.0119572\pi\)
−0.999295 + 0.0375558i \(0.988043\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 0 0
\(713\) − 16.0000i − 0.599205i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 2.00000i 0.0746914i
\(718\) 0 0
\(719\) 20.0000 0.745874 0.372937 0.927857i \(-0.378351\pi\)
0.372937 + 0.927857i \(0.378351\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 18.0000i 0.669427i
\(724\) 0 0
\(725\) 66.0000 2.45118
\(726\) 0 0
\(727\) 24.0000 0.890111 0.445055 0.895503i \(-0.353184\pi\)
0.445055 + 0.895503i \(0.353184\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 8.00000 0.295891
\(732\) 0 0
\(733\) − 2.00000i − 0.0738717i −0.999318 0.0369358i \(-0.988240\pi\)
0.999318 0.0369358i \(-0.0117597\pi\)
\(734\) 0 0
\(735\) 28.0000i 1.03280i
\(736\) 0 0
\(737\) 8.00000 0.294684
\(738\) 0 0
\(739\) − 28.0000i − 1.03000i −0.857191 0.514998i \(-0.827793\pi\)
0.857191 0.514998i \(-0.172207\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 34.0000i − 1.24734i −0.781688 0.623670i \(-0.785641\pi\)
0.781688 0.623670i \(-0.214359\pi\)
\(744\) 0 0
\(745\) 64.0000 2.34478
\(746\) 0 0
\(747\) − 14.0000i − 0.512233i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 48.0000 1.75154 0.875772 0.482724i \(-0.160353\pi\)
0.875772 + 0.482724i \(0.160353\pi\)
\(752\) 0 0
\(753\) 8.00000 0.291536
\(754\) 0 0
\(755\) 80.0000 2.91150
\(756\) 0 0
\(757\) −22.0000 −0.799604 −0.399802 0.916602i \(-0.630921\pi\)
−0.399802 + 0.916602i \(0.630921\pi\)
\(758\) 0 0
\(759\) − 8.00000i − 0.290382i
\(760\) 0 0
\(761\) 20.0000i 0.724999i 0.931984 + 0.362500i \(0.118077\pi\)
−0.931984 + 0.362500i \(0.881923\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 8.00000i 0.289241i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) − 26.0000i − 0.937584i −0.883309 0.468792i \(-0.844689\pi\)
0.883309 0.468792i \(-0.155311\pi\)
\(770\) 0 0
\(771\) 14.0000 0.504198
\(772\) 0 0
\(773\) 48.0000i 1.72644i 0.504828 + 0.863220i \(0.331556\pi\)
−0.504828 + 0.863220i \(0.668444\pi\)
\(774\) 0 0
\(775\) − 44.0000i − 1.58053i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 96.0000 3.43956
\(780\) 0 0
\(781\) −4.00000 −0.143131
\(782\) 0 0
\(783\) 6.00000 0.214423
\(784\) 0 0
\(785\) 8.00000i 0.285532i
\(786\) 0 0
\(787\) − 44.0000i − 1.56843i −0.620489 0.784215i \(-0.713066\pi\)
0.620489 0.784215i \(-0.286934\pi\)
\(788\) 0 0
\(789\) 20.0000 0.712019
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) − 8.00000i − 0.283731i
\(796\) 0 0
\(797\) 2.00000 0.0708436 0.0354218 0.999372i \(-0.488723\pi\)
0.0354218 + 0.999372i \(0.488723\pi\)
\(798\) 0 0
\(799\) 12.0000i 0.424529i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −4.00000 −0.141157
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −26.0000 −0.915243
\(808\) 0 0
\(809\) −10.0000 −0.351581 −0.175791 0.984428i \(-0.556248\pi\)
−0.175791 + 0.984428i \(0.556248\pi\)
\(810\) 0 0
\(811\) 8.00000i 0.280918i 0.990086 + 0.140459i \(0.0448578\pi\)
−0.990086 + 0.140459i \(0.955142\pi\)
\(812\) 0 0
\(813\) − 8.00000i − 0.280572i
\(814\) 0 0
\(815\) 96.0000 3.36273
\(816\) 0 0
\(817\) 32.0000i 1.11954i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 24.0000i − 0.837606i −0.908077 0.418803i \(-0.862450\pi\)
0.908077 0.418803i \(-0.137550\pi\)
\(822\) 0 0
\(823\) 16.0000 0.557725 0.278862 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) 0 0
\(825\) − 22.0000i − 0.765942i
\(826\) 0 0
\(827\) − 10.0000i − 0.347734i −0.984769 0.173867i \(-0.944374\pi\)
0.984769 0.173867i \(-0.0556263\pi\)
\(828\) 0 0
\(829\) −34.0000 −1.18087 −0.590434 0.807086i \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) 0 0
\(831\) −2.00000 −0.0693792
\(832\) 0 0
\(833\) −14.0000 −0.485071
\(834\) 0 0
\(835\) 72.0000 2.49166
\(836\) 0 0
\(837\) − 4.00000i − 0.138260i
\(838\) 0 0
\(839\) 30.0000i 1.03572i 0.855467 + 0.517858i \(0.173270\pi\)
−0.855467 + 0.517858i \(0.826730\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) − 20.0000i − 0.688837i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 20.0000 0.686398
\(850\) 0 0
\(851\) − 24.0000i − 0.822709i
\(852\) 0 0
\(853\) 38.0000i 1.30110i 0.759465 + 0.650548i \(0.225461\pi\)
−0.759465 + 0.650548i \(0.774539\pi\)
\(854\) 0 0
\(855\) −32.0000 −1.09438
\(856\) 0 0
\(857\) −30.0000 −1.02478 −0.512390 0.858753i \(-0.671240\pi\)
−0.512390 + 0.858753i \(0.671240\pi\)
\(858\) 0 0
\(859\) −36.0000 −1.22830 −0.614152 0.789188i \(-0.710502\pi\)
−0.614152 + 0.789188i \(0.710502\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 2.00000i 0.0680808i 0.999420 + 0.0340404i \(0.0108375\pi\)
−0.999420 + 0.0340404i \(0.989163\pi\)
\(864\) 0 0
\(865\) 72.0000i 2.44807i
\(866\) 0 0
\(867\) 13.0000 0.441503
\(868\) 0 0
\(869\) 16.0000i 0.542763i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 10.0000i 0.338449i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 42.0000i 1.41824i 0.705088 + 0.709120i \(0.250907\pi\)
−0.705088 + 0.709120i \(0.749093\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −38.0000 −1.28025 −0.640126 0.768270i \(-0.721118\pi\)
−0.640126 + 0.768270i \(0.721118\pi\)
\(882\) 0 0
\(883\) −44.0000 −1.48072 −0.740359 0.672212i \(-0.765344\pi\)
−0.740359 + 0.672212i \(0.765344\pi\)
\(884\) 0 0
\(885\) 56.0000 1.88242
\(886\) 0 0
\(887\) 32.0000 1.07445 0.537227 0.843437i \(-0.319472\pi\)
0.537227 + 0.843437i \(0.319472\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) − 2.00000i − 0.0670025i
\(892\) 0 0
\(893\) −48.0000 −1.60626
\(894\) 0 0
\(895\) − 48.0000i − 1.60446i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 24.0000i − 0.800445i
\(900\) 0 0
\(901\) 4.00000 0.133259
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 24.0000i 0.797787i
\(906\) 0 0
\(907\) 20.0000 0.664089 0.332045 0.943264i \(-0.392262\pi\)
0.332045 + 0.943264i \(0.392262\pi\)
\(908\) 0 0
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) 16.0000 0.530104 0.265052 0.964234i \(-0.414611\pi\)
0.265052 + 0.964234i \(0.414611\pi\)
\(912\) 0 0
\(913\) −28.0000 −0.926665
\(914\) 0 0
\(915\) 40.0000i 1.32236i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) 12.0000i 0.395413i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) − 66.0000i − 2.17007i
\(926\) 0 0
\(927\) 8.00000 0.262754
\(928\) 0 0
\(929\) − 20.0000i − 0.656179i −0.944647 0.328089i \(-0.893595\pi\)
0.944647 0.328089i \(-0.106405\pi\)
\(930\) 0 0
\(931\) − 56.0000i − 1.83533i
\(932\) 0 0
\(933\) −12.0000 −0.392862
\(934\) 0 0
\(935\) 16.0000 0.523256
\(936\) 0 0
\(937\) −18.0000 −0.588034 −0.294017 0.955800i \(-0.594992\pi\)
−0.294017 + 0.955800i \(0.594992\pi\)
\(938\) 0 0
\(939\) −6.00000 −0.195803
\(940\) 0 0
\(941\) − 48.0000i − 1.56476i −0.622804 0.782378i \(-0.714007\pi\)
0.622804 0.782378i \(-0.285993\pi\)
\(942\) 0 0
\(943\) − 48.0000i − 1.56310i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 42.0000i 1.36482i 0.730971 + 0.682408i \(0.239067\pi\)
−0.730971 + 0.682408i \(0.760933\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) − 12.0000i − 0.389127i
\(952\) 0 0
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 12.0000i − 0.387905i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 15.0000 0.483871
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 104.000 3.34788
\(966\) 0 0
\(967\) 28.0000i 0.900419i 0.892923 + 0.450210i \(0.148651\pi\)
−0.892923 + 0.450210i \(0.851349\pi\)
\(968\) 0 0
\(969\) − 16.0000i − 0.513994i
\(970\) 0 0
\(971\) −60.0000 −1.92549 −0.962746 0.270408i \(-0.912841\pi\)
−0.962746 + 0.270408i \(0.912841\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) − 10.0000i − 0.319275i
\(982\) 0 0
\(983\) 30.0000i 0.956851i 0.878128 + 0.478426i \(0.158792\pi\)
−0.878128 + 0.478426i \(0.841208\pi\)
\(984\) 0 0
\(985\) −48.0000 −1.52941
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 16.0000 0.508770
\(990\) 0 0
\(991\) −32.0000 −1.01651 −0.508257 0.861206i \(-0.669710\pi\)
−0.508257 + 0.861206i \(0.669710\pi\)
\(992\) 0 0
\(993\) − 28.0000i − 0.888553i
\(994\) 0 0
\(995\) − 32.0000i − 1.01447i
\(996\) 0 0
\(997\) −26.0000 −0.823428 −0.411714 0.911313i \(-0.635070\pi\)
−0.411714 + 0.911313i \(0.635070\pi\)
\(998\) 0 0
\(999\) − 6.00000i − 0.189832i
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 4056.2.c.d.337.1 2
13.5 odd 4 4056.2.a.a.1.1 1
13.8 odd 4 312.2.a.c.1.1 1
13.12 even 2 inner 4056.2.c.d.337.2 2
39.8 even 4 936.2.a.a.1.1 1
52.31 even 4 8112.2.a.q.1.1 1
52.47 even 4 624.2.a.j.1.1 1
65.34 odd 4 7800.2.a.s.1.1 1
104.21 odd 4 2496.2.a.p.1.1 1
104.99 even 4 2496.2.a.a.1.1 1
156.47 odd 4 1872.2.a.b.1.1 1
312.125 even 4 7488.2.a.cb.1.1 1
312.203 odd 4 7488.2.a.cc.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
312.2.a.c.1.1 1 13.8 odd 4
624.2.a.j.1.1 1 52.47 even 4
936.2.a.a.1.1 1 39.8 even 4
1872.2.a.b.1.1 1 156.47 odd 4
2496.2.a.a.1.1 1 104.99 even 4
2496.2.a.p.1.1 1 104.21 odd 4
4056.2.a.a.1.1 1 13.5 odd 4
4056.2.c.d.337.1 2 1.1 even 1 trivial
4056.2.c.d.337.2 2 13.12 even 2 inner
7488.2.a.cb.1.1 1 312.125 even 4
7488.2.a.cc.1.1 1 312.203 odd 4
7800.2.a.s.1.1 1 65.34 odd 4
8112.2.a.q.1.1 1 52.31 even 4