Properties

Label 7056.2.k.g.881.8
Level $7056$
Weight $2$
Character 7056.881
Analytic conductor $56.342$
Analytic rank $0$
Dimension $8$
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.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(56.3424436662\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
Defining polynomial: \(x^{8} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 441)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 881.8
Root \(0.382683 + 0.923880i\) of defining polynomial
Character \(\chi\) \(=\) 7056.881
Dual form 7056.2.k.g.881.7

$q$-expansion

\(f(q)\) \(=\) \(q+3.37849 q^{5} +O(q^{10})\) \(q+3.37849 q^{5} +0.828427i q^{11} -3.37849i q^{13} +1.39942 q^{17} +6.75699i q^{19} +2.00000i q^{23} +6.41421 q^{25} +4.82843i q^{29} -6.75699i q^{31} -2.58579 q^{37} +8.15640 q^{41} +12.4853 q^{43} -6.75699 q^{47} +7.07107i q^{53} +2.79884i q^{55} -6.75699 q^{59} +8.15640i q^{61} -11.4142i q^{65} -8.48528 q^{67} +4.82843i q^{71} +1.39942i q^{73} +9.65685 q^{79} +13.5140 q^{83} +4.72792 q^{85} +6.17733 q^{89} +22.8284i q^{95} -1.39942i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + O(q^{10}) \) \( 8 q + 40 q^{25} - 32 q^{37} + 32 q^{43} + 32 q^{79} - 64 q^{85} + O(q^{100}) \)

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
\(5\) 3.37849 1.51091 0.755454 0.655202i \(-0.227416\pi\)
0.755454 + 0.655202i \(0.227416\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.828427i 0.249780i 0.992171 + 0.124890i \(0.0398578\pi\)
−0.992171 + 0.124890i \(0.960142\pi\)
\(12\) 0 0
\(13\) − 3.37849i − 0.937025i −0.883457 0.468513i \(-0.844790\pi\)
0.883457 0.468513i \(-0.155210\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.39942 0.339409 0.169704 0.985495i \(-0.445719\pi\)
0.169704 + 0.985495i \(0.445719\pi\)
\(18\) 0 0
\(19\) 6.75699i 1.55016i 0.631864 + 0.775079i \(0.282290\pi\)
−0.631864 + 0.775079i \(0.717710\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.00000i 0.417029i 0.978019 + 0.208514i \(0.0668628\pi\)
−0.978019 + 0.208514i \(0.933137\pi\)
\(24\) 0 0
\(25\) 6.41421 1.28284
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.82843i 0.896616i 0.893879 + 0.448308i \(0.147973\pi\)
−0.893879 + 0.448308i \(0.852027\pi\)
\(30\) 0 0
\(31\) − 6.75699i − 1.21359i −0.794858 0.606795i \(-0.792455\pi\)
0.794858 0.606795i \(-0.207545\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.58579 −0.425101 −0.212550 0.977150i \(-0.568177\pi\)
−0.212550 + 0.977150i \(0.568177\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 8.15640 1.27382 0.636908 0.770940i \(-0.280213\pi\)
0.636908 + 0.770940i \(0.280213\pi\)
\(42\) 0 0
\(43\) 12.4853 1.90399 0.951994 0.306117i \(-0.0990300\pi\)
0.951994 + 0.306117i \(0.0990300\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −6.75699 −0.985608 −0.492804 0.870140i \(-0.664028\pi\)
−0.492804 + 0.870140i \(0.664028\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 7.07107i 0.971286i 0.874157 + 0.485643i \(0.161414\pi\)
−0.874157 + 0.485643i \(0.838586\pi\)
\(54\) 0 0
\(55\) 2.79884i 0.377395i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −6.75699 −0.879685 −0.439842 0.898075i \(-0.644966\pi\)
−0.439842 + 0.898075i \(0.644966\pi\)
\(60\) 0 0
\(61\) 8.15640i 1.04432i 0.852847 + 0.522160i \(0.174874\pi\)
−0.852847 + 0.522160i \(0.825126\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 11.4142i − 1.41576i
\(66\) 0 0
\(67\) −8.48528 −1.03664 −0.518321 0.855186i \(-0.673443\pi\)
−0.518321 + 0.855186i \(0.673443\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 4.82843i 0.573029i 0.958076 + 0.286514i \(0.0924966\pi\)
−0.958076 + 0.286514i \(0.907503\pi\)
\(72\) 0 0
\(73\) 1.39942i 0.163789i 0.996641 + 0.0818947i \(0.0260971\pi\)
−0.996641 + 0.0818947i \(0.973903\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 9.65685 1.08648 0.543240 0.839577i \(-0.317197\pi\)
0.543240 + 0.839577i \(0.317197\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 13.5140 1.48335 0.741676 0.670759i \(-0.234031\pi\)
0.741676 + 0.670759i \(0.234031\pi\)
\(84\) 0 0
\(85\) 4.72792 0.512815
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.17733 0.654795 0.327398 0.944887i \(-0.393828\pi\)
0.327398 + 0.944887i \(0.393828\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 22.8284i 2.34215i
\(96\) 0 0
\(97\) − 1.39942i − 0.142089i −0.997473 0.0710447i \(-0.977367\pi\)
0.997473 0.0710447i \(-0.0226333\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 12.9343 1.28701 0.643506 0.765441i \(-0.277479\pi\)
0.643506 + 0.765441i \(0.277479\pi\)
\(102\) 0 0
\(103\) 2.79884i 0.275777i 0.990448 + 0.137889i \(0.0440316\pi\)
−0.990448 + 0.137889i \(0.955968\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 9.31371i − 0.900390i −0.892930 0.450195i \(-0.851354\pi\)
0.892930 0.450195i \(-0.148646\pi\)
\(108\) 0 0
\(109\) −2.58579 −0.247673 −0.123837 0.992303i \(-0.539520\pi\)
−0.123837 + 0.992303i \(0.539520\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 5.89949i 0.554978i 0.960729 + 0.277489i \(0.0895022\pi\)
−0.960729 + 0.277489i \(0.910498\pi\)
\(114\) 0 0
\(115\) 6.75699i 0.630092i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 10.3137 0.937610
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 4.77791 0.427349
\(126\) 0 0
\(127\) 0.485281 0.0430618 0.0215309 0.999768i \(-0.493146\pi\)
0.0215309 + 0.999768i \(0.493146\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 9.55582 0.834896 0.417448 0.908701i \(-0.362925\pi\)
0.417448 + 0.908701i \(0.362925\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0.828427i 0.0707773i 0.999374 + 0.0353887i \(0.0112669\pi\)
−0.999374 + 0.0353887i \(0.988733\pi\)
\(138\) 0 0
\(139\) − 9.55582i − 0.810514i −0.914203 0.405257i \(-0.867182\pi\)
0.914203 0.405257i \(-0.132818\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2.79884 0.234050
\(144\) 0 0
\(145\) 16.3128i 1.35470i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 4.24264i − 0.347571i −0.984784 0.173785i \(-0.944400\pi\)
0.984784 0.173785i \(-0.0555999\pi\)
\(150\) 0 0
\(151\) 12.4853 1.01604 0.508019 0.861346i \(-0.330378\pi\)
0.508019 + 0.861346i \(0.330378\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 22.8284i − 1.83362i
\(156\) 0 0
\(157\) 7.33664i 0.585528i 0.956185 + 0.292764i \(0.0945750\pi\)
−0.956185 + 0.292764i \(0.905425\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −14.8284 −1.16145 −0.580726 0.814099i \(-0.697231\pi\)
−0.580726 + 0.814099i \(0.697231\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2.79884 −0.216580 −0.108290 0.994119i \(-0.534538\pi\)
−0.108290 + 0.994119i \(0.534538\pi\)
\(168\) 0 0
\(169\) 1.58579 0.121984
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 8.15640 0.620120 0.310060 0.950717i \(-0.399651\pi\)
0.310060 + 0.950717i \(0.399651\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 9.51472i − 0.711163i −0.934645 0.355582i \(-0.884283\pi\)
0.934645 0.355582i \(-0.115717\pi\)
\(180\) 0 0
\(181\) 17.7122i 1.31654i 0.752782 + 0.658270i \(0.228711\pi\)
−0.752782 + 0.658270i \(0.771289\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −8.73606 −0.642288
\(186\) 0 0
\(187\) 1.15932i 0.0847775i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 20.8284i − 1.50709i −0.657395 0.753546i \(-0.728342\pi\)
0.657395 0.753546i \(-0.271658\pi\)
\(192\) 0 0
\(193\) −17.6569 −1.27097 −0.635484 0.772114i \(-0.719199\pi\)
−0.635484 + 0.772114i \(0.719199\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 16.2426i − 1.15724i −0.815597 0.578620i \(-0.803591\pi\)
0.815597 0.578620i \(-0.196409\pi\)
\(198\) 0 0
\(199\) − 19.1116i − 1.35479i −0.735620 0.677394i \(-0.763109\pi\)
0.735620 0.677394i \(-0.236891\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 27.5563 1.92462
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −5.59767 −0.387199
\(210\) 0 0
\(211\) −7.31371 −0.503496 −0.251748 0.967793i \(-0.581005\pi\)
−0.251748 + 0.967793i \(0.581005\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 42.1814 2.87675
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 4.72792i − 0.318034i
\(222\) 0 0
\(223\) 23.0698i 1.54487i 0.635095 + 0.772434i \(0.280961\pi\)
−0.635095 + 0.772434i \(0.719039\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −2.79884 −0.185765 −0.0928826 0.995677i \(-0.529608\pi\)
−0.0928826 + 0.995677i \(0.529608\pi\)
\(228\) 0 0
\(229\) 12.1146i 0.800552i 0.916395 + 0.400276i \(0.131086\pi\)
−0.916395 + 0.400276i \(0.868914\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 10.8284i 0.709394i 0.934981 + 0.354697i \(0.115416\pi\)
−0.934981 + 0.354697i \(0.884584\pi\)
\(234\) 0 0
\(235\) −22.8284 −1.48916
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 0.343146i − 0.0221963i −0.999938 0.0110981i \(-0.996467\pi\)
0.999938 0.0110981i \(-0.00353272\pi\)
\(240\) 0 0
\(241\) − 8.97616i − 0.578205i −0.957298 0.289103i \(-0.906643\pi\)
0.957298 0.289103i \(-0.0933569\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 22.8284 1.45254
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 25.8686 1.63281 0.816407 0.577477i \(-0.195963\pi\)
0.816407 + 0.577477i \(0.195963\pi\)
\(252\) 0 0
\(253\) −1.65685 −0.104166
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 21.6704 1.35176 0.675880 0.737011i \(-0.263764\pi\)
0.675880 + 0.737011i \(0.263764\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 24.1421i 1.48867i 0.667808 + 0.744334i \(0.267233\pi\)
−0.667808 + 0.744334i \(0.732767\pi\)
\(264\) 0 0
\(265\) 23.8896i 1.46752i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −31.2262 −1.90389 −0.951947 0.306262i \(-0.900922\pi\)
−0.951947 + 0.306262i \(0.900922\pi\)
\(270\) 0 0
\(271\) − 6.75699i − 0.410458i −0.978714 0.205229i \(-0.934206\pi\)
0.978714 0.205229i \(-0.0657939\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 5.31371i 0.320429i
\(276\) 0 0
\(277\) −20.9706 −1.26000 −0.630000 0.776596i \(-0.716945\pi\)
−0.630000 + 0.776596i \(0.716945\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 18.8284i − 1.12321i −0.827406 0.561605i \(-0.810184\pi\)
0.827406 0.561605i \(-0.189816\pi\)
\(282\) 0 0
\(283\) 29.8268i 1.77302i 0.462711 + 0.886509i \(0.346877\pi\)
−0.462711 + 0.886509i \(0.653123\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −15.0416 −0.884802
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 7.33664 0.428611 0.214306 0.976767i \(-0.431251\pi\)
0.214306 + 0.976767i \(0.431251\pi\)
\(294\) 0 0
\(295\) −22.8284 −1.32912
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 6.75699 0.390767
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 27.5563i 1.57787i
\(306\) 0 0
\(307\) 12.3547i 0.705117i 0.935790 + 0.352559i \(0.114688\pi\)
−0.935790 + 0.352559i \(0.885312\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −10.7151 −0.607600 −0.303800 0.952736i \(-0.598255\pi\)
−0.303800 + 0.952736i \(0.598255\pi\)
\(312\) 0 0
\(313\) − 12.9343i − 0.731091i −0.930794 0.365545i \(-0.880883\pi\)
0.930794 0.365545i \(-0.119117\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 23.7574i − 1.33435i −0.744903 0.667173i \(-0.767504\pi\)
0.744903 0.667173i \(-0.232496\pi\)
\(318\) 0 0
\(319\) −4.00000 −0.223957
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 9.45584i 0.526137i
\(324\) 0 0
\(325\) − 21.6704i − 1.20206i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −10.3431 −0.568511 −0.284255 0.958749i \(-0.591746\pi\)
−0.284255 + 0.958749i \(0.591746\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −28.6675 −1.56627
\(336\) 0 0
\(337\) −12.9289 −0.704284 −0.352142 0.935947i \(-0.614547\pi\)
−0.352142 + 0.935947i \(0.614547\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 5.59767 0.303131
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 18.9706i − 1.01839i −0.860650 0.509197i \(-0.829943\pi\)
0.860650 0.509197i \(-0.170057\pi\)
\(348\) 0 0
\(349\) 6.17733i 0.330665i 0.986238 + 0.165332i \(0.0528697\pi\)
−0.986238 + 0.165332i \(0.947130\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 23.6494 1.25873 0.629367 0.777109i \(-0.283314\pi\)
0.629367 + 0.777109i \(0.283314\pi\)
\(354\) 0 0
\(355\) 16.3128i 0.865794i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 1.02944i − 0.0543316i −0.999631 0.0271658i \(-0.991352\pi\)
0.999631 0.0271658i \(-0.00864821\pi\)
\(360\) 0 0
\(361\) −26.6569 −1.40299
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 4.72792i 0.247471i
\(366\) 0 0
\(367\) 3.95815i 0.206614i 0.994650 + 0.103307i \(0.0329424\pi\)
−0.994650 + 0.103307i \(0.967058\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −22.6274 −1.17160 −0.585802 0.810454i \(-0.699220\pi\)
−0.585802 + 0.810454i \(0.699220\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 16.3128 0.840152
\(378\) 0 0
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 27.0279 1.38106 0.690532 0.723302i \(-0.257377\pi\)
0.690532 + 0.723302i \(0.257377\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 13.7990i 0.699637i 0.936818 + 0.349818i \(0.113757\pi\)
−0.936818 + 0.349818i \(0.886243\pi\)
\(390\) 0 0
\(391\) 2.79884i 0.141543i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 32.6256 1.64157
\(396\) 0 0
\(397\) − 27.2680i − 1.36854i −0.729227 0.684272i \(-0.760120\pi\)
0.729227 0.684272i \(-0.239880\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 10.8284i − 0.540746i −0.962756 0.270373i \(-0.912853\pi\)
0.962756 0.270373i \(-0.0871470\pi\)
\(402\) 0 0
\(403\) −22.8284 −1.13716
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 2.14214i − 0.106182i
\(408\) 0 0
\(409\) 17.7122i 0.875813i 0.899020 + 0.437907i \(0.144280\pi\)
−0.899020 + 0.437907i \(0.855720\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 45.6569 2.24121
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 35.4244 1.73060 0.865299 0.501256i \(-0.167129\pi\)
0.865299 + 0.501256i \(0.167129\pi\)
\(420\) 0 0
\(421\) −23.3137 −1.13624 −0.568120 0.822946i \(-0.692329\pi\)
−0.568120 + 0.822946i \(0.692329\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 8.97616 0.435408
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 15.4558i − 0.744482i −0.928136 0.372241i \(-0.878590\pi\)
0.928136 0.372241i \(-0.121410\pi\)
\(432\) 0 0
\(433\) − 8.15640i − 0.391972i −0.980607 0.195986i \(-0.937209\pi\)
0.980607 0.195986i \(-0.0627907\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −13.5140 −0.646461
\(438\) 0 0
\(439\) 28.6675i 1.36822i 0.729377 + 0.684112i \(0.239810\pi\)
−0.729377 + 0.684112i \(0.760190\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 24.8284i − 1.17963i −0.807537 0.589817i \(-0.799200\pi\)
0.807537 0.589817i \(-0.200800\pi\)
\(444\) 0 0
\(445\) 20.8701 0.989336
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 24.2426i − 1.14408i −0.820225 0.572040i \(-0.806152\pi\)
0.820225 0.572040i \(-0.193848\pi\)
\(450\) 0 0
\(451\) 6.75699i 0.318174i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 28.6274 1.33913 0.669567 0.742752i \(-0.266480\pi\)
0.669567 + 0.742752i \(0.266480\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 21.6704 1.00929 0.504645 0.863327i \(-0.331623\pi\)
0.504645 + 0.863327i \(0.331623\pi\)
\(462\) 0 0
\(463\) −3.51472 −0.163343 −0.0816714 0.996659i \(-0.526026\pi\)
−0.0816714 + 0.996659i \(0.526026\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 2.79884 0.129515 0.0647573 0.997901i \(-0.479373\pi\)
0.0647573 + 0.997901i \(0.479373\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 10.3431i 0.475578i
\(474\) 0 0
\(475\) 43.3407i 1.98861i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 8.39651 0.383646 0.191823 0.981430i \(-0.438560\pi\)
0.191823 + 0.981430i \(0.438560\pi\)
\(480\) 0 0
\(481\) 8.73606i 0.398330i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 4.72792i − 0.214684i
\(486\) 0 0
\(487\) −9.45584 −0.428485 −0.214243 0.976780i \(-0.568728\pi\)
−0.214243 + 0.976780i \(0.568728\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 2.68629i − 0.121231i −0.998161 0.0606153i \(-0.980694\pi\)
0.998161 0.0606153i \(-0.0193063\pi\)
\(492\) 0 0
\(493\) 6.75699i 0.304319i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 2.14214 0.0958952 0.0479476 0.998850i \(-0.484732\pi\)
0.0479476 + 0.998850i \(0.484732\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −32.6256 −1.45470 −0.727352 0.686264i \(-0.759249\pi\)
−0.727352 + 0.686264i \(0.759249\pi\)
\(504\) 0 0
\(505\) 43.6985 1.94456
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 36.8239 1.63219 0.816095 0.577918i \(-0.196135\pi\)
0.816095 + 0.577918i \(0.196135\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 9.45584i 0.416674i
\(516\) 0 0
\(517\) − 5.59767i − 0.246185i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −42.7611 −1.87340 −0.936699 0.350136i \(-0.886135\pi\)
−0.936699 + 0.350136i \(0.886135\pi\)
\(522\) 0 0
\(523\) 17.4721i 0.764003i 0.924162 + 0.382001i \(0.124765\pi\)
−0.924162 + 0.382001i \(0.875235\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 9.45584i − 0.411903i
\(528\) 0 0
\(529\) 19.0000 0.826087
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 27.5563i − 1.19360i
\(534\) 0 0
\(535\) − 31.4663i − 1.36041i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 18.9706 0.815608 0.407804 0.913069i \(-0.366295\pi\)
0.407804 + 0.913069i \(0.366295\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −8.73606 −0.374212
\(546\) 0 0
\(547\) 26.6274 1.13851 0.569253 0.822162i \(-0.307232\pi\)
0.569253 + 0.822162i \(0.307232\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −32.6256 −1.38990
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 10.5858i 0.448534i 0.974528 + 0.224267i \(0.0719988\pi\)
−0.974528 + 0.224267i \(0.928001\pi\)
\(558\) 0 0
\(559\) − 42.1814i − 1.78408i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −2.79884 −0.117957 −0.0589784 0.998259i \(-0.518784\pi\)
−0.0589784 + 0.998259i \(0.518784\pi\)
\(564\) 0 0
\(565\) 19.9314i 0.838520i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 25.1127i − 1.05278i −0.850244 0.526390i \(-0.823545\pi\)
0.850244 0.526390i \(-0.176455\pi\)
\(570\) 0 0
\(571\) 11.3137 0.473464 0.236732 0.971575i \(-0.423924\pi\)
0.236732 + 0.971575i \(0.423924\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 12.8284i 0.534982i
\(576\) 0 0
\(577\) − 13.7541i − 0.572590i −0.958142 0.286295i \(-0.907576\pi\)
0.958142 0.286295i \(-0.0924237\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −5.85786 −0.242608
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 20.2710 0.836672 0.418336 0.908292i \(-0.362613\pi\)
0.418336 + 0.908292i \(0.362613\pi\)
\(588\) 0 0
\(589\) 45.6569 1.88126
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −20.5111 −0.842288 −0.421144 0.906994i \(-0.638371\pi\)
−0.421144 + 0.906994i \(0.638371\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 6.48528i 0.264981i 0.991184 + 0.132491i \(0.0422975\pi\)
−0.991184 + 0.132491i \(0.957703\pi\)
\(600\) 0 0
\(601\) − 27.6076i − 1.12614i −0.826410 0.563069i \(-0.809621\pi\)
0.826410 0.563069i \(-0.190379\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 34.8448 1.41664
\(606\) 0 0
\(607\) 23.0698i 0.936374i 0.883629 + 0.468187i \(0.155093\pi\)
−0.883629 + 0.468187i \(0.844907\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 22.8284i 0.923539i
\(612\) 0 0
\(613\) −10.8701 −0.439037 −0.219519 0.975608i \(-0.570449\pi\)
−0.219519 + 0.975608i \(0.570449\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 30.4853i 1.22729i 0.789582 + 0.613646i \(0.210298\pi\)
−0.789582 + 0.613646i \(0.789702\pi\)
\(618\) 0 0
\(619\) 24.7093i 0.993151i 0.867994 + 0.496576i \(0.165409\pi\)
−0.867994 + 0.496576i \(0.834591\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −15.9289 −0.637157
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −3.61859 −0.144283
\(630\) 0 0
\(631\) 21.1716 0.842827 0.421414 0.906869i \(-0.361534\pi\)
0.421414 + 0.906869i \(0.361534\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1.63952 0.0650624
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 7.51472i − 0.296814i −0.988926 0.148407i \(-0.952586\pi\)
0.988926 0.148407i \(-0.0474145\pi\)
\(642\) 0 0
\(643\) 6.75699i 0.266469i 0.991085 + 0.133235i \(0.0425364\pi\)
−0.991085 + 0.133235i \(0.957464\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −25.8686 −1.01700 −0.508500 0.861062i \(-0.669800\pi\)
−0.508500 + 0.861062i \(0.669800\pi\)
\(648\) 0 0
\(649\) − 5.59767i − 0.219728i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 38.4853i 1.50605i 0.657995 + 0.753023i \(0.271405\pi\)
−0.657995 + 0.753023i \(0.728595\pi\)
\(654\) 0 0
\(655\) 32.2843 1.26145
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 47.4558i − 1.84862i −0.381646 0.924309i \(-0.624643\pi\)
0.381646 0.924309i \(-0.375357\pi\)
\(660\) 0 0
\(661\) − 7.33664i − 0.285362i −0.989769 0.142681i \(-0.954428\pi\)
0.989769 0.142681i \(-0.0455724\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −9.65685 −0.373915
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −6.75699 −0.260851
\(672\) 0 0
\(673\) −37.8995 −1.46092 −0.730459 0.682956i \(-0.760694\pi\)
−0.730459 + 0.682956i \(0.760694\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −32.8657 −1.26313 −0.631566 0.775322i \(-0.717588\pi\)
−0.631566 + 0.775322i \(0.717588\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 38.7696i 1.48348i 0.670690 + 0.741738i \(0.265998\pi\)
−0.670690 + 0.741738i \(0.734002\pi\)
\(684\) 0 0
\(685\) 2.79884i 0.106938i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 23.8896 0.910119
\(690\) 0 0
\(691\) − 9.55582i − 0.363521i −0.983343 0.181760i \(-0.941821\pi\)
0.983343 0.181760i \(-0.0581795\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 32.2843i − 1.22461i
\(696\) 0 0
\(697\) 11.4142 0.432344
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 40.7696i 1.53984i 0.638138 + 0.769922i \(0.279705\pi\)
−0.638138 + 0.769922i \(0.720295\pi\)
\(702\) 0 0
\(703\) − 17.4721i − 0.658974i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −32.7279 −1.22912 −0.614561 0.788869i \(-0.710667\pi\)
−0.614561 + 0.788869i \(0.710667\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 13.5140 0.506102
\(714\) 0 0
\(715\) 9.45584 0.353629
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 13.5140 0.503986 0.251993 0.967729i \(-0.418914\pi\)
0.251993 + 0.967729i \(0.418914\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 30.9706i 1.15022i
\(726\) 0 0
\(727\) − 43.3407i − 1.60742i −0.595022 0.803710i \(-0.702857\pi\)
0.595022 0.803710i \(-0.297143\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 17.4721 0.646230
\(732\) 0 0
\(733\) 23.3099i 0.860971i 0.902598 + 0.430485i \(0.141658\pi\)
−0.902598 + 0.430485i \(0.858342\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 7.02944i − 0.258933i
\(738\) 0 0
\(739\) 18.8284 0.692615 0.346307 0.938121i \(-0.387435\pi\)
0.346307 + 0.938121i \(0.387435\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 52.4264i 1.92334i 0.274212 + 0.961669i \(0.411583\pi\)
−0.274212 + 0.961669i \(0.588417\pi\)
\(744\) 0 0
\(745\) − 14.3337i − 0.525147i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −20.6863 −0.754853 −0.377427 0.926039i \(-0.623191\pi\)
−0.377427 + 0.926039i \(0.623191\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 42.1814 1.53514
\(756\) 0 0
\(757\) 6.87006 0.249696 0.124848 0.992176i \(-0.460156\pi\)
0.124848 + 0.992176i \(0.460156\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −0.579658 −0.0210126 −0.0105063 0.999945i \(-0.503344\pi\)
−0.0105063 + 0.999945i \(0.503344\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 22.8284i 0.824287i
\(768\) 0 0
\(769\) 45.8995i 1.65518i 0.561335 + 0.827589i \(0.310288\pi\)
−0.561335 + 0.827589i \(0.689712\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 18.8715 0.678762 0.339381 0.940649i \(-0.389782\pi\)
0.339381 + 0.940649i \(0.389782\pi\)
\(774\) 0 0
\(775\) − 43.3407i − 1.55685i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 55.1127i 1.97462i
\(780\) 0 0
\(781\) −4.00000 −0.143131
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 24.7868i 0.884679i
\(786\) 0 0
\(787\) − 13.5140i − 0.481721i −0.970560 0.240861i \(-0.922570\pi\)
0.970560 0.240861i \(-0.0774296\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 27.5563 0.978555
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.73897 −0.0615976 −0.0307988 0.999526i \(-0.509805\pi\)
−0.0307988 + 0.999526i \(0.509805\pi\)
\(798\) 0 0
\(799\) −9.45584 −0.334524
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −1.15932 −0.0409114
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 32.0416i − 1.12652i −0.826278 0.563262i \(-0.809546\pi\)
0.826278 0.563262i \(-0.190454\pi\)
\(810\) 0 0
\(811\) 9.55582i 0.335550i 0.985825 + 0.167775i \(0.0536583\pi\)
−0.985825 + 0.167775i \(0.946342\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −50.0977 −1.75485
\(816\) 0 0
\(817\) 84.3629i 2.95148i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 32.2426i − 1.12528i −0.826703 0.562638i \(-0.809787\pi\)
0.826703 0.562638i \(-0.190213\pi\)
\(822\) 0 0
\(823\) −25.9411 −0.904251 −0.452125 0.891954i \(-0.649334\pi\)
−0.452125 + 0.891954i \(0.649334\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 28.3431i 0.985588i 0.870146 + 0.492794i \(0.164024\pi\)
−0.870146 + 0.492794i \(0.835976\pi\)
\(828\) 0 0
\(829\) 34.8448i 1.21021i 0.796146 + 0.605105i \(0.206869\pi\)
−0.796146 + 0.605105i \(0.793131\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −9.45584 −0.327233
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 20.2710 0.699831 0.349916 0.936781i \(-0.386210\pi\)
0.349916 + 0.936781i \(0.386210\pi\)
\(840\) 0 0
\(841\) 5.68629 0.196079
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 5.35757 0.184306
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 5.17157i − 0.177279i
\(852\) 0 0
\(853\) − 38.8029i − 1.32859i −0.747472 0.664294i \(-0.768732\pi\)
0.747472 0.664294i \(-0.231268\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −19.6913 −0.672642 −0.336321 0.941747i \(-0.609183\pi\)
−0.336321 + 0.941747i \(0.609183\pi\)
\(858\) 0 0
\(859\) − 29.8268i − 1.01768i −0.860862 0.508838i \(-0.830075\pi\)
0.860862 0.508838i \(-0.169925\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 33.3137i − 1.13401i −0.823714 0.567006i \(-0.808102\pi\)
0.823714 0.567006i \(-0.191898\pi\)
\(864\) 0 0
\(865\) 27.5563 0.936944
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 8.00000i 0.271381i
\(870\) 0 0
\(871\) 28.6675i 0.971360i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 32.5269 1.09836 0.549178 0.835705i \(-0.314941\pi\)
0.549178 + 0.835705i \(0.314941\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 20.5111 0.691035 0.345518 0.938412i \(-0.387703\pi\)
0.345518 + 0.938412i \(0.387703\pi\)
\(882\) 0 0
\(883\) 37.4558 1.26049 0.630245 0.776397i \(-0.282955\pi\)
0.630245 + 0.776397i \(0.282955\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −17.9523 −0.602780 −0.301390 0.953501i \(-0.597451\pi\)
−0.301390 + 0.953501i \(0.597451\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 45.6569i − 1.52785i
\(894\) 0 0
\(895\) − 32.1454i − 1.07450i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 32.6256 1.08813
\(900\) 0 0
\(901\) 9.89538i 0.329663i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 59.8406i 1.98917i
\(906\) 0 0
\(907\) 44.7696 1.48655 0.743274 0.668987i \(-0.233272\pi\)
0.743274 + 0.668987i \(0.233272\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 25.5147i 0.845340i 0.906284 + 0.422670i \(0.138907\pi\)
−0.906284 + 0.422670i \(0.861093\pi\)
\(912\) 0 0
\(913\) 11.1953i 0.370512i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −9.45584 −0.311920 −0.155960 0.987763i \(-0.549847\pi\)
−0.155960 + 0.987763i \(0.549847\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 16.3128 0.536943
\(924\) 0 0
\(925\) −16.5858 −0.545337
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −19.6913 −0.646051 −0.323025 0.946390i \(-0.604700\pi\)
−0.323025 + 0.946390i \(0.604700\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 3.91674i 0.128091i
\(936\) 0 0
\(937\) − 1.73897i − 0.0568098i −0.999596 0.0284049i \(-0.990957\pi\)
0.999596 0.0284049i \(-0.00904277\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 14.9134 0.486163 0.243081 0.970006i \(-0.421842\pi\)
0.243081 + 0.970006i \(0.421842\pi\)
\(942\) 0 0
\(943\) 16.3128i 0.531218i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 22.9706i 0.746443i 0.927742 + 0.373221i \(0.121747\pi\)
−0.927742 + 0.373221i \(0.878253\pi\)
\(948\) 0 0
\(949\) 4.72792 0.153475
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 26.3848i 0.854687i 0.904089 + 0.427343i \(0.140550\pi\)
−0.904089 + 0.427343i \(0.859450\pi\)
\(954\) 0 0
\(955\) − 70.3687i − 2.27708i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −14.6569 −0.472802
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −59.6536 −1.92032
\(966\) 0 0
\(967\) −43.1127 −1.38641 −0.693205 0.720740i \(-0.743802\pi\)
−0.693205 + 0.720740i \(0.743802\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −32.6256 −1.04701 −0.523503 0.852024i \(-0.675375\pi\)
−0.523503 + 0.852024i \(0.675375\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 39.1716i − 1.25321i −0.779337 0.626605i \(-0.784444\pi\)
0.779337 0.626605i \(-0.215556\pi\)
\(978\) 0 0
\(979\) 5.11747i 0.163555i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −17.4721 −0.557274 −0.278637 0.960396i \(-0.589883\pi\)
−0.278637 + 0.960396i \(0.589883\pi\)
\(984\) 0 0
\(985\) − 54.8756i − 1.74848i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 24.9706i 0.794018i
\(990\) 0 0
\(991\) −59.3970 −1.88681 −0.943403 0.331647i \(-0.892396\pi\)
−0.943403 + 0.331647i \(0.892396\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 64.5685i − 2.04696i
\(996\) 0 0
\(997\) − 11.7750i − 0.372918i −0.982463 0.186459i \(-0.940299\pi\)
0.982463 0.186459i \(-0.0597011\pi\)
\(998\) 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.k.g.881.8 8
3.2 odd 2 inner 7056.2.k.g.881.1 8
4.3 odd 2 441.2.c.b.440.2 yes 8
7.6 odd 2 inner 7056.2.k.g.881.2 8
12.11 even 2 441.2.c.b.440.7 yes 8
21.20 even 2 inner 7056.2.k.g.881.7 8
28.3 even 6 441.2.p.c.215.8 16
28.11 odd 6 441.2.p.c.215.7 16
28.19 even 6 441.2.p.c.80.2 16
28.23 odd 6 441.2.p.c.80.1 16
28.27 even 2 441.2.c.b.440.1 8
84.11 even 6 441.2.p.c.215.2 16
84.23 even 6 441.2.p.c.80.8 16
84.47 odd 6 441.2.p.c.80.7 16
84.59 odd 6 441.2.p.c.215.1 16
84.83 odd 2 441.2.c.b.440.8 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
441.2.c.b.440.1 8 28.27 even 2
441.2.c.b.440.2 yes 8 4.3 odd 2
441.2.c.b.440.7 yes 8 12.11 even 2
441.2.c.b.440.8 yes 8 84.83 odd 2
441.2.p.c.80.1 16 28.23 odd 6
441.2.p.c.80.2 16 28.19 even 6
441.2.p.c.80.7 16 84.47 odd 6
441.2.p.c.80.8 16 84.23 even 6
441.2.p.c.215.1 16 84.59 odd 6
441.2.p.c.215.2 16 84.11 even 6
441.2.p.c.215.7 16 28.11 odd 6
441.2.p.c.215.8 16 28.3 even 6
7056.2.k.g.881.1 8 3.2 odd 2 inner
7056.2.k.g.881.2 8 7.6 odd 2 inner
7056.2.k.g.881.7 8 21.20 even 2 inner
7056.2.k.g.881.8 8 1.1 even 1 trivial