Properties

Label 7056.2.b.v.1567.3
Level $7056$
Weight $2$
Character 7056.1567
Analytic conductor $56.342$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 7056 = 2^{4} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7056.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(56.3424436662\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{5})\)
Defining polynomial: \(x^{4} - x^{3} + 2 x^{2} + x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
Twist minimal: no (minimal twist has level 1008)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1567.3
Root \(0.809017 + 1.40126i\) of defining polynomial
Character \(\chi\) \(=\) 7056.1567
Dual form 7056.2.b.v.1567.2

$q$-expansion

\(f(q)\) \(=\) \(q+3.87298i q^{5} +O(q^{10})\) \(q+3.87298i q^{5} +3.87298i q^{11} -3.46410i q^{13} +4.00000 q^{19} +7.74597i q^{23} -10.0000 q^{25} -6.70820 q^{29} -1.00000 q^{31} +4.00000 q^{37} +7.74597i q^{41} -6.92820i q^{43} -13.4164 q^{47} +6.70820 q^{53} -15.0000 q^{55} -6.70820 q^{59} +10.3923i q^{61} +13.4164 q^{65} -6.92820i q^{67} -7.74597i q^{71} +6.92820i q^{73} -12.1244i q^{79} -6.70820 q^{83} +7.74597i q^{89} +15.4919i q^{95} +5.19615i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 16q^{19} - 40q^{25} - 4q^{31} + 16q^{37} - 60q^{55} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7056\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1765\) \(4609\) \(6175\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 3.87298i 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.87298i 1.16775i 0.811844 + 0.583874i \(0.198464\pi\)
−0.811844 + 0.583874i \(0.801536\pi\)
\(12\) 0 0
\(13\) − 3.46410i − 0.960769i −0.877058 0.480384i \(-0.840497\pi\)
0.877058 0.480384i \(-0.159503\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 7.74597i 1.61515i 0.589768 + 0.807573i \(0.299219\pi\)
−0.589768 + 0.807573i \(0.700781\pi\)
\(24\) 0 0
\(25\) −10.0000 −2.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −6.70820 −1.24568 −0.622841 0.782348i \(-0.714022\pi\)
−0.622841 + 0.782348i \(0.714022\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605 −0.0898027 0.995960i \(-0.528624\pi\)
−0.0898027 + 0.995960i \(0.528624\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 7.74597i 1.20972i 0.796333 + 0.604858i \(0.206770\pi\)
−0.796333 + 0.604858i \(0.793230\pi\)
\(42\) 0 0
\(43\) − 6.92820i − 1.05654i −0.849076 0.528271i \(-0.822841\pi\)
0.849076 0.528271i \(-0.177159\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −13.4164 −1.95698 −0.978492 0.206284i \(-0.933863\pi\)
−0.978492 + 0.206284i \(0.933863\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.70820 0.921443 0.460721 0.887545i \(-0.347591\pi\)
0.460721 + 0.887545i \(0.347591\pi\)
\(54\) 0 0
\(55\) −15.0000 −2.02260
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −6.70820 −0.873334 −0.436667 0.899623i \(-0.643841\pi\)
−0.436667 + 0.899623i \(0.643841\pi\)
\(60\) 0 0
\(61\) 10.3923i 1.33060i 0.746577 + 0.665299i \(0.231696\pi\)
−0.746577 + 0.665299i \(0.768304\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 13.4164 1.66410
\(66\) 0 0
\(67\) − 6.92820i − 0.846415i −0.906033 0.423207i \(-0.860904\pi\)
0.906033 0.423207i \(-0.139096\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 7.74597i − 0.919277i −0.888106 0.459639i \(-0.847979\pi\)
0.888106 0.459639i \(-0.152021\pi\)
\(72\) 0 0
\(73\) 6.92820i 0.810885i 0.914121 + 0.405442i \(0.132883\pi\)
−0.914121 + 0.405442i \(0.867117\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) − 12.1244i − 1.36410i −0.731307 0.682048i \(-0.761089\pi\)
0.731307 0.682048i \(-0.238911\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −6.70820 −0.736321 −0.368161 0.929762i \(-0.620012\pi\)
−0.368161 + 0.929762i \(0.620012\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 7.74597i 0.821071i 0.911845 + 0.410535i \(0.134658\pi\)
−0.911845 + 0.410535i \(0.865342\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 15.4919i 1.58944i
\(96\) 0 0
\(97\) 5.19615i 0.527589i 0.964579 + 0.263795i \(0.0849741\pi\)
−0.964579 + 0.263795i \(0.915026\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 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
\(107\) 11.6190i 1.12325i 0.827393 + 0.561623i \(0.189823\pi\)
−0.827393 + 0.561623i \(0.810177\pi\)
\(108\) 0 0
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 13.4164 1.26211 0.631055 0.775738i \(-0.282622\pi\)
0.631055 + 0.775738i \(0.282622\pi\)
\(114\) 0 0
\(115\) −30.0000 −2.79751
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −4.00000 −0.363636
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 19.3649i − 1.73205i
\(126\) 0 0
\(127\) 12.1244i 1.07586i 0.842989 + 0.537931i \(0.180794\pi\)
−0.842989 + 0.537931i \(0.819206\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −6.70820 −0.586098 −0.293049 0.956097i \(-0.594670\pi\)
−0.293049 + 0.956097i \(0.594670\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) 2.00000 0.169638 0.0848189 0.996396i \(-0.472969\pi\)
0.0848189 + 0.996396i \(0.472969\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 13.4164 1.12194
\(144\) 0 0
\(145\) − 25.9808i − 2.15758i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −13.4164 −1.09911 −0.549557 0.835456i \(-0.685204\pi\)
−0.549557 + 0.835456i \(0.685204\pi\)
\(150\) 0 0
\(151\) − 15.5885i − 1.26857i −0.773099 0.634285i \(-0.781294\pi\)
0.773099 0.634285i \(-0.218706\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 3.87298i − 0.311086i
\(156\) 0 0
\(157\) 20.7846i 1.65879i 0.558661 + 0.829396i \(0.311315\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 10.3923i 0.813988i 0.913431 + 0.406994i \(0.133423\pi\)
−0.913431 + 0.406994i \(0.866577\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 15.4919i − 1.17783i −0.808195 0.588915i \(-0.799555\pi\)
0.808195 0.588915i \(-0.200445\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 23.2379i − 1.73688i −0.495792 0.868441i \(-0.665122\pi\)
0.495792 0.868441i \(-0.334878\pi\)
\(180\) 0 0
\(181\) 24.2487i 1.80239i 0.433411 + 0.901196i \(0.357310\pi\)
−0.433411 + 0.901196i \(0.642690\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 15.4919i 1.13899i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 15.4919i − 1.12096i −0.828169 0.560478i \(-0.810617\pi\)
0.828169 0.560478i \(-0.189383\pi\)
\(192\) 0 0
\(193\) 7.00000 0.503871 0.251936 0.967744i \(-0.418933\pi\)
0.251936 + 0.967744i \(0.418933\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −13.4164 −0.955879 −0.477940 0.878393i \(-0.658616\pi\)
−0.477940 + 0.878393i \(0.658616\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −30.0000 −2.09529
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 15.4919i 1.07160i
\(210\) 0 0
\(211\) − 10.3923i − 0.715436i −0.933830 0.357718i \(-0.883555\pi\)
0.933830 0.357718i \(-0.116445\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 26.8328 1.82998
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −23.0000 −1.54019 −0.770097 0.637927i \(-0.779792\pi\)
−0.770097 + 0.637927i \(0.779792\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −6.70820 −0.445239 −0.222620 0.974905i \(-0.571461\pi\)
−0.222620 + 0.974905i \(0.571461\pi\)
\(228\) 0 0
\(229\) − 17.3205i − 1.14457i −0.820054 0.572286i \(-0.806057\pi\)
0.820054 0.572286i \(-0.193943\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) − 51.9615i − 3.38960i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 23.2379i 1.50313i 0.659656 + 0.751567i \(0.270702\pi\)
−0.659656 + 0.751567i \(0.729298\pi\)
\(240\) 0 0
\(241\) − 15.5885i − 1.00414i −0.864827 0.502070i \(-0.832572\pi\)
0.864827 0.502070i \(-0.167428\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 13.8564i − 0.881662i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 6.70820 0.423418 0.211709 0.977333i \(-0.432097\pi\)
0.211709 + 0.977333i \(0.432097\pi\)
\(252\) 0 0
\(253\) −30.0000 −1.88608
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 7.74597i − 0.483180i −0.970378 0.241590i \(-0.922331\pi\)
0.970378 0.241590i \(-0.0776689\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 15.4919i − 0.955274i −0.878557 0.477637i \(-0.841494\pi\)
0.878557 0.477637i \(-0.158506\pi\)
\(264\) 0 0
\(265\) 25.9808i 1.59599i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 27.1109i − 1.65298i −0.562952 0.826490i \(-0.690334\pi\)
0.562952 0.826490i \(-0.309666\pi\)
\(270\) 0 0
\(271\) −7.00000 −0.425220 −0.212610 0.977137i \(-0.568196\pi\)
−0.212610 + 0.977137i \(0.568196\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 38.7298i − 2.33550i
\(276\) 0 0
\(277\) −20.0000 −1.20168 −0.600842 0.799368i \(-0.705168\pi\)
−0.600842 + 0.799368i \(0.705168\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 26.8328 1.60071 0.800356 0.599525i \(-0.204644\pi\)
0.800356 + 0.599525i \(0.204644\pi\)
\(282\) 0 0
\(283\) −20.0000 −1.18888 −0.594438 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 17.0000 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 11.6190i − 0.678786i −0.940645 0.339393i \(-0.889778\pi\)
0.940645 0.339393i \(-0.110222\pi\)
\(294\) 0 0
\(295\) − 25.9808i − 1.51266i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 26.8328 1.55178
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −40.2492 −2.30466
\(306\) 0 0
\(307\) −14.0000 −0.799022 −0.399511 0.916728i \(-0.630820\pi\)
−0.399511 + 0.916728i \(0.630820\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −13.4164 −0.760775 −0.380387 0.924827i \(-0.624209\pi\)
−0.380387 + 0.924827i \(0.624209\pi\)
\(312\) 0 0
\(313\) 1.73205i 0.0979013i 0.998801 + 0.0489506i \(0.0155877\pi\)
−0.998801 + 0.0489506i \(0.984412\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −20.1246 −1.13031 −0.565155 0.824984i \(-0.691184\pi\)
−0.565155 + 0.824984i \(0.691184\pi\)
\(318\) 0 0
\(319\) − 25.9808i − 1.45464i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 34.6410i 1.92154i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 6.92820i 0.380808i 0.981706 + 0.190404i \(0.0609799\pi\)
−0.981706 + 0.190404i \(0.939020\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 26.8328 1.46603
\(336\) 0 0
\(337\) −1.00000 −0.0544735 −0.0272367 0.999629i \(-0.508671\pi\)
−0.0272367 + 0.999629i \(0.508671\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 3.87298i − 0.209734i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 7.74597i 0.415825i 0.978147 + 0.207913i \(0.0666670\pi\)
−0.978147 + 0.207913i \(0.933333\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 15.4919i − 0.824552i −0.911059 0.412276i \(-0.864734\pi\)
0.911059 0.412276i \(-0.135266\pi\)
\(354\) 0 0
\(355\) 30.0000 1.59223
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −26.8328 −1.40449
\(366\) 0 0
\(367\) −11.0000 −0.574195 −0.287098 0.957901i \(-0.592690\pi\)
−0.287098 + 0.957901i \(0.592690\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 23.2379i 1.19681i
\(378\) 0 0
\(379\) 17.3205i 0.889695i 0.895606 + 0.444847i \(0.146742\pi\)
−0.895606 + 0.444847i \(0.853258\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 13.4164 0.685546 0.342773 0.939418i \(-0.388634\pi\)
0.342773 + 0.939418i \(0.388634\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 13.4164 0.680239 0.340119 0.940382i \(-0.389532\pi\)
0.340119 + 0.940382i \(0.389532\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 46.9574 2.36268
\(396\) 0 0
\(397\) 13.8564i 0.695433i 0.937600 + 0.347717i \(0.113043\pi\)
−0.937600 + 0.347717i \(0.886957\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 3.46410i 0.172559i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 15.4919i 0.767907i
\(408\) 0 0
\(409\) 8.66025i 0.428222i 0.976809 + 0.214111i \(0.0686854\pi\)
−0.976809 + 0.214111i \(0.931315\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) − 25.9808i − 1.27535i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 34.0000 1.65706 0.828529 0.559946i \(-0.189178\pi\)
0.828529 + 0.559946i \(0.189178\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 38.7298i − 1.86555i −0.360459 0.932775i \(-0.617380\pi\)
0.360459 0.932775i \(-0.382620\pi\)
\(432\) 0 0
\(433\) − 6.92820i − 0.332948i −0.986046 0.166474i \(-0.946762\pi\)
0.986046 0.166474i \(-0.0532382\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 30.9839i 1.48216i
\(438\) 0 0
\(439\) 37.0000 1.76591 0.882957 0.469454i \(-0.155549\pi\)
0.882957 + 0.469454i \(0.155549\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 3.87298i 0.184011i 0.995758 + 0.0920055i \(0.0293277\pi\)
−0.995758 + 0.0920055i \(0.970672\pi\)
\(444\) 0 0
\(445\) −30.0000 −1.42214
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −13.4164 −0.633159 −0.316580 0.948566i \(-0.602534\pi\)
−0.316580 + 0.948566i \(0.602534\pi\)
\(450\) 0 0
\(451\) −30.0000 −1.41264
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 29.0000 1.35656 0.678281 0.734802i \(-0.262725\pi\)
0.678281 + 0.734802i \(0.262725\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 15.4919i − 0.721531i −0.932657 0.360766i \(-0.882515\pi\)
0.932657 0.360766i \(-0.117485\pi\)
\(462\) 0 0
\(463\) 24.2487i 1.12693i 0.826139 + 0.563467i \(0.190533\pi\)
−0.826139 + 0.563467i \(0.809467\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 26.8328 1.23377
\(474\) 0 0
\(475\) −40.0000 −1.83533
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −26.8328 −1.22602 −0.613011 0.790074i \(-0.710042\pi\)
−0.613011 + 0.790074i \(0.710042\pi\)
\(480\) 0 0
\(481\) − 13.8564i − 0.631798i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −20.1246 −0.913812
\(486\) 0 0
\(487\) 12.1244i 0.549407i 0.961529 + 0.274703i \(0.0885797\pi\)
−0.961529 + 0.274703i \(0.911420\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 19.3649i 0.873926i 0.899479 + 0.436963i \(0.143946\pi\)
−0.899479 + 0.436963i \(0.856054\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) − 17.3205i − 0.775372i −0.921791 0.387686i \(-0.873274\pi\)
0.921791 0.387686i \(-0.126726\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −26.8328 −1.19642 −0.598208 0.801341i \(-0.704120\pi\)
−0.598208 + 0.801341i \(0.704120\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 11.6190i 0.515001i 0.966278 + 0.257500i \(0.0828989\pi\)
−0.966278 + 0.257500i \(0.917101\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 30.9839i 1.36531i
\(516\) 0 0
\(517\) − 51.9615i − 2.28527i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 15.4919i 0.678714i 0.940658 + 0.339357i \(0.110209\pi\)
−0.940658 + 0.339357i \(0.889791\pi\)
\(522\) 0 0
\(523\) −22.0000 −0.961993 −0.480996 0.876723i \(-0.659725\pi\)
−0.480996 + 0.876723i \(0.659725\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −37.0000 −1.60870
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 26.8328 1.16226
\(534\) 0 0
\(535\) −45.0000 −1.94552
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −32.0000 −1.37579 −0.687894 0.725811i \(-0.741464\pi\)
−0.687894 + 0.725811i \(0.741464\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 15.4919i − 0.663602i
\(546\) 0 0
\(547\) − 41.5692i − 1.77737i −0.458517 0.888686i \(-0.651619\pi\)
0.458517 0.888686i \(-0.348381\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −26.8328 −1.14312
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −33.5410 −1.42118 −0.710589 0.703607i \(-0.751572\pi\)
−0.710589 + 0.703607i \(0.751572\pi\)
\(558\) 0 0
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −33.5410 −1.41359 −0.706793 0.707421i \(-0.749859\pi\)
−0.706793 + 0.707421i \(0.749859\pi\)
\(564\) 0 0
\(565\) 51.9615i 2.18604i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 26.8328 1.12489 0.562445 0.826835i \(-0.309861\pi\)
0.562445 + 0.826835i \(0.309861\pi\)
\(570\) 0 0
\(571\) 17.3205i 0.724841i 0.932015 + 0.362420i \(0.118050\pi\)
−0.932015 + 0.362420i \(0.881950\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 77.4597i − 3.23029i
\(576\) 0 0
\(577\) 22.5167i 0.937381i 0.883362 + 0.468690i \(0.155274\pi\)
−0.883362 + 0.468690i \(0.844726\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 25.9808i 1.07601i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −6.70820 −0.276877 −0.138439 0.990371i \(-0.544208\pi\)
−0.138439 + 0.990371i \(0.544208\pi\)
\(588\) 0 0
\(589\) −4.00000 −0.164817
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 7.74597i − 0.318089i −0.987271 0.159044i \(-0.949159\pi\)
0.987271 0.159044i \(-0.0508413\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 30.9839i 1.26597i 0.774165 + 0.632983i \(0.218170\pi\)
−0.774165 + 0.632983i \(0.781830\pi\)
\(600\) 0 0
\(601\) 43.3013i 1.76630i 0.469095 + 0.883148i \(0.344580\pi\)
−0.469095 + 0.883148i \(0.655420\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 15.4919i − 0.629837i
\(606\) 0 0
\(607\) −11.0000 −0.446476 −0.223238 0.974764i \(-0.571663\pi\)
−0.223238 + 0.974764i \(0.571663\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 46.4758i 1.88021i
\(612\) 0 0
\(613\) −10.0000 −0.403896 −0.201948 0.979396i \(-0.564727\pi\)
−0.201948 + 0.979396i \(0.564727\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 40.2492 1.62037 0.810186 0.586172i \(-0.199366\pi\)
0.810186 + 0.586172i \(0.199366\pi\)
\(618\) 0 0
\(619\) 38.0000 1.52735 0.763674 0.645601i \(-0.223393\pi\)
0.763674 + 0.645601i \(0.223393\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) − 36.3731i − 1.44799i −0.689806 0.723994i \(-0.742304\pi\)
0.689806 0.723994i \(-0.257696\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −46.9574 −1.86345
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 13.4164 0.529916 0.264958 0.964260i \(-0.414642\pi\)
0.264958 + 0.964260i \(0.414642\pi\)
\(642\) 0 0
\(643\) −8.00000 −0.315489 −0.157745 0.987480i \(-0.550422\pi\)
−0.157745 + 0.987480i \(0.550422\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 40.2492 1.58236 0.791180 0.611583i \(-0.209467\pi\)
0.791180 + 0.611583i \(0.209467\pi\)
\(648\) 0 0
\(649\) − 25.9808i − 1.01983i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 46.9574 1.83759 0.918793 0.394740i \(-0.129165\pi\)
0.918793 + 0.394740i \(0.129165\pi\)
\(654\) 0 0
\(655\) − 25.9808i − 1.01515i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 7.74597i − 0.301740i −0.988554 0.150870i \(-0.951793\pi\)
0.988554 0.150870i \(-0.0482075\pi\)
\(660\) 0 0
\(661\) 6.92820i 0.269476i 0.990881 + 0.134738i \(0.0430193\pi\)
−0.990881 + 0.134738i \(0.956981\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 51.9615i − 2.01196i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −40.2492 −1.55380
\(672\) 0 0
\(673\) 13.0000 0.501113 0.250557 0.968102i \(-0.419386\pi\)
0.250557 + 0.968102i \(0.419386\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 3.87298i 0.148851i 0.997227 + 0.0744254i \(0.0237123\pi\)
−0.997227 + 0.0744254i \(0.976288\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 3.87298i 0.148196i 0.997251 + 0.0740978i \(0.0236077\pi\)
−0.997251 + 0.0740978i \(0.976392\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 23.2379i − 0.885293i
\(690\) 0 0
\(691\) 14.0000 0.532585 0.266293 0.963892i \(-0.414201\pi\)
0.266293 + 0.963892i \(0.414201\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 7.74597i 0.293821i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −20.1246 −0.760096 −0.380048 0.924967i \(-0.624093\pi\)
−0.380048 + 0.924967i \(0.624093\pi\)
\(702\) 0 0
\(703\) 16.0000 0.603451
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −22.0000 −0.826227 −0.413114 0.910679i \(-0.635559\pi\)
−0.413114 + 0.910679i \(0.635559\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 7.74597i − 0.290089i
\(714\) 0 0
\(715\) 51.9615i 1.94325i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −40.2492 −1.50104 −0.750521 0.660846i \(-0.770198\pi\)
−0.750521 + 0.660846i \(0.770198\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 67.0820 2.49136
\(726\) 0 0
\(727\) 5.00000 0.185440 0.0927199 0.995692i \(-0.470444\pi\)
0.0927199 + 0.995692i \(0.470444\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 6.92820i 0.255899i 0.991781 + 0.127950i \(0.0408395\pi\)
−0.991781 + 0.127950i \(0.959160\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 26.8328 0.988399
\(738\) 0 0
\(739\) − 3.46410i − 0.127429i −0.997968 0.0637145i \(-0.979705\pi\)
0.997968 0.0637145i \(-0.0202947\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 15.4919i 0.568344i 0.958773 + 0.284172i \(0.0917187\pi\)
−0.958773 + 0.284172i \(0.908281\pi\)
\(744\) 0 0
\(745\) − 51.9615i − 1.90372i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 50.2295i 1.83290i 0.400150 + 0.916450i \(0.368958\pi\)
−0.400150 + 0.916450i \(0.631042\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 60.3738 2.19723
\(756\) 0 0
\(757\) −34.0000 −1.23575 −0.617876 0.786276i \(-0.712006\pi\)
−0.617876 + 0.786276i \(0.712006\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 23.2379i 0.842373i 0.906974 + 0.421187i \(0.138386\pi\)
−0.906974 + 0.421187i \(0.861614\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 23.2379i 0.839072i
\(768\) 0 0
\(769\) − 12.1244i − 0.437215i −0.975813 0.218608i \(-0.929848\pi\)
0.975813 0.218608i \(-0.0701515\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 15.4919i 0.557206i 0.960406 + 0.278603i \(0.0898714\pi\)
−0.960406 + 0.278603i \(0.910129\pi\)
\(774\) 0 0
\(775\) 10.0000 0.359211
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 30.9839i 1.11011i
\(780\) 0 0
\(781\) 30.0000 1.07348
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −80.4984 −2.87311
\(786\) 0 0
\(787\) 14.0000 0.499046 0.249523 0.968369i \(-0.419726\pi\)
0.249523 + 0.968369i \(0.419726\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 36.0000 1.27840
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 50.3488i 1.78345i 0.452582 + 0.891723i \(0.350503\pi\)
−0.452582 + 0.891723i \(0.649497\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −26.8328 −0.946910
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 26.8328 0.943392 0.471696 0.881761i \(-0.343642\pi\)
0.471696 + 0.881761i \(0.343642\pi\)
\(810\) 0 0
\(811\) 44.0000 1.54505 0.772524 0.634985i \(-0.218994\pi\)
0.772524 + 0.634985i \(0.218994\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −40.2492 −1.40987
\(816\) 0 0
\(817\) − 27.7128i − 0.969549i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 6.70820 0.234118 0.117059 0.993125i \(-0.462653\pi\)
0.117059 + 0.993125i \(0.462653\pi\)
\(822\) 0 0
\(823\) − 10.3923i − 0.362253i −0.983460 0.181126i \(-0.942026\pi\)
0.983460 0.181126i \(-0.0579743\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 3.87298i − 0.134677i −0.997730 0.0673384i \(-0.978549\pi\)
0.997730 0.0673384i \(-0.0214507\pi\)
\(828\) 0 0
\(829\) 13.8564i 0.481253i 0.970618 + 0.240626i \(0.0773529\pi\)
−0.970618 + 0.240626i \(0.922647\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −26.8328 −0.926372 −0.463186 0.886261i \(-0.653294\pi\)
−0.463186 + 0.886261i \(0.653294\pi\)
\(840\) 0 0
\(841\) 16.0000 0.551724
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 3.87298i 0.133235i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 30.9839i 1.06211i
\(852\) 0 0
\(853\) 6.92820i 0.237217i 0.992941 + 0.118609i \(0.0378434\pi\)
−0.992941 + 0.118609i \(0.962157\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 54.2218i 1.85218i 0.377302 + 0.926090i \(0.376852\pi\)
−0.377302 + 0.926090i \(0.623148\pi\)
\(858\) 0 0
\(859\) 16.0000 0.545913 0.272956 0.962026i \(-0.411998\pi\)
0.272956 + 0.962026i \(0.411998\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 23.2379i 0.791027i 0.918460 + 0.395514i \(0.129433\pi\)
−0.918460 + 0.395514i \(0.870567\pi\)
\(864\) 0 0
\(865\) 60.0000 2.04006
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 46.9574 1.59292
\(870\) 0 0
\(871\) −24.0000 −0.813209
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −4.00000 −0.135070 −0.0675352 0.997717i \(-0.521513\pi\)
−0.0675352 + 0.997717i \(0.521513\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 15.4919i − 0.521936i −0.965347 0.260968i \(-0.915958\pi\)
0.965347 0.260968i \(-0.0840418\pi\)
\(882\) 0 0
\(883\) − 3.46410i − 0.116576i −0.998300 0.0582882i \(-0.981436\pi\)
0.998300 0.0582882i \(-0.0185642\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 13.4164 0.450479 0.225239 0.974303i \(-0.427684\pi\)
0.225239 + 0.974303i \(0.427684\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −53.6656 −1.79585
\(894\) 0 0
\(895\) 90.0000 3.00837
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 6.70820 0.223731
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −93.9149 −3.12184
\(906\) 0 0
\(907\) − 3.46410i − 0.115024i −0.998345 0.0575118i \(-0.981683\pi\)
0.998345 0.0575118i \(-0.0183167\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 23.2379i 0.769906i 0.922936 + 0.384953i \(0.125782\pi\)
−0.922936 + 0.384953i \(0.874218\pi\)
\(912\) 0 0
\(913\) − 25.9808i − 0.859838i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 38.1051i 1.25697i 0.777821 + 0.628486i \(0.216325\pi\)
−0.777821 + 0.628486i \(0.783675\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −26.8328 −0.883213
\(924\) 0 0
\(925\) −40.0000 −1.31519
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 46.4758i 1.52482i 0.647093 + 0.762411i \(0.275984\pi\)
−0.647093 + 0.762411i \(0.724016\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 12.1244i 0.396085i 0.980193 + 0.198043i \(0.0634585\pi\)
−0.980193 + 0.198043i \(0.936542\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 19.3649i − 0.631278i −0.948879 0.315639i \(-0.897781\pi\)
0.948879 0.315639i \(-0.102219\pi\)
\(942\) 0 0
\(943\) −60.0000 −1.95387
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 7.74597i 0.251710i 0.992049 + 0.125855i \(0.0401674\pi\)
−0.992049 + 0.125855i \(0.959833\pi\)
\(948\) 0 0
\(949\) 24.0000 0.779073
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −26.8328 −0.869200 −0.434600 0.900624i \(-0.643110\pi\)
−0.434600 + 0.900624i \(0.643110\pi\)
\(954\) 0 0
\(955\) 60.0000 1.94155
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −30.0000 −0.967742
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 27.1109i 0.872730i
\(966\) 0 0
\(967\) 29.4449i 0.946883i 0.880825 + 0.473441i \(0.156988\pi\)
−0.880825 + 0.473441i \(0.843012\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 46.9574 1.50694 0.753468 0.657485i \(-0.228380\pi\)
0.753468 + 0.657485i \(0.228380\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −53.6656 −1.71692 −0.858458 0.512884i \(-0.828577\pi\)
−0.858458 + 0.512884i \(0.828577\pi\)
\(978\) 0 0
\(979\) −30.0000 −0.958804
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 13.4164 0.427917 0.213958 0.976843i \(-0.431364\pi\)
0.213958 + 0.976843i \(0.431364\pi\)
\(984\) 0 0
\(985\) − 51.9615i − 1.65563i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 53.6656 1.70647
\(990\) 0 0
\(991\) − 1.73205i − 0.0550204i −0.999622 0.0275102i \(-0.991242\pi\)
0.999622 0.0275102i \(-0.00875787\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 30.9839i − 0.982255i
\(996\) 0 0
\(997\) 55.4256i 1.75535i 0.479259 + 0.877674i \(0.340906\pi\)
−0.479259 + 0.877674i \(0.659094\pi\)
\(998\) 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 7056.2.b.v.1567.3 4
3.2 odd 2 inner 7056.2.b.v.1567.1 4
4.3 odd 2 7056.2.b.o.1567.3 4
7.2 even 3 1008.2.cs.o.703.2 yes 4
7.3 odd 6 1008.2.cs.p.271.2 yes 4
7.6 odd 2 7056.2.b.o.1567.2 4
12.11 even 2 7056.2.b.o.1567.1 4
21.2 odd 6 1008.2.cs.o.703.1 yes 4
21.17 even 6 1008.2.cs.p.271.1 yes 4
21.20 even 2 7056.2.b.o.1567.4 4
28.3 even 6 1008.2.cs.o.271.2 yes 4
28.23 odd 6 1008.2.cs.p.703.2 yes 4
28.27 even 2 inner 7056.2.b.v.1567.2 4
84.23 even 6 1008.2.cs.p.703.1 yes 4
84.59 odd 6 1008.2.cs.o.271.1 4
84.83 odd 2 inner 7056.2.b.v.1567.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1008.2.cs.o.271.1 4 84.59 odd 6
1008.2.cs.o.271.2 yes 4 28.3 even 6
1008.2.cs.o.703.1 yes 4 21.2 odd 6
1008.2.cs.o.703.2 yes 4 7.2 even 3
1008.2.cs.p.271.1 yes 4 21.17 even 6
1008.2.cs.p.271.2 yes 4 7.3 odd 6
1008.2.cs.p.703.1 yes 4 84.23 even 6
1008.2.cs.p.703.2 yes 4 28.23 odd 6
7056.2.b.o.1567.1 4 12.11 even 2
7056.2.b.o.1567.2 4 7.6 odd 2
7056.2.b.o.1567.3 4 4.3 odd 2
7056.2.b.o.1567.4 4 21.20 even 2
7056.2.b.v.1567.1 4 3.2 odd 2 inner
7056.2.b.v.1567.2 4 28.27 even 2 inner
7056.2.b.v.1567.3 4 1.1 even 1 trivial
7056.2.b.v.1567.4 4 84.83 odd 2 inner