Properties

Label 7056.2.b.w.1567.8
Level $7056$
Weight $2$
Character 7056.1567
Analytic conductor $56.342$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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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: \(8\)
Coefficient field: 8.0.339738624.1
Defining polynomial: \(x^{8} - 4 x^{6} + 14 x^{4} - 8 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 2352)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1567.8
Root \(-0.662827 + 0.382683i\) of defining polynomial
Character \(\chi\) \(=\) 7056.1567
Dual form 7056.2.b.w.1567.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.29725i q^{5} +O(q^{10})\) \(q+4.29725i q^{5} +5.06262i q^{11} -3.37849i q^{13} +2.76652i q^{17} +4.70319 q^{19} -4.83120i q^{23} -13.4663 q^{25} -2.46054 q^{29} -5.69764 q^{31} -2.33530 q^{37} -5.14822i q^{41} +13.0199i q^{43} -5.35449 q^{47} +4.22371 q^{53} -21.7553 q^{55} -9.60498 q^{59} +3.87698i q^{61} +14.5182 q^{65} +4.76342i q^{67} +12.3181i q^{71} -11.5102i q^{73} -13.9166i q^{79} -7.32191 q^{83} -11.8884 q^{85} -14.0448i q^{89} +20.2108i q^{95} +14.5716i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 24q^{25} - 16q^{29} - 32q^{31} + 16q^{47} - 16q^{53} - 64q^{55} - 48q^{59} + 16q^{65} - 64q^{85} + O(q^{100}) \)

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
\(5\) 4.29725i 1.92179i 0.276916 + 0.960894i \(0.410688\pi\)
−0.276916 + 0.960894i \(0.589312\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.06262i 1.52644i 0.646141 + 0.763218i \(0.276382\pi\)
−0.646141 + 0.763218i \(0.723618\pi\)
\(12\) 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
\(17\) 2.76652i 0.670978i 0.942044 + 0.335489i \(0.108902\pi\)
−0.942044 + 0.335489i \(0.891098\pi\)
\(18\) 0 0
\(19\) 4.70319 1.07898 0.539492 0.841990i \(-0.318616\pi\)
0.539492 + 0.841990i \(0.318616\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 4.83120i − 1.00737i −0.863886 0.503687i \(-0.831976\pi\)
0.863886 0.503687i \(-0.168024\pi\)
\(24\) 0 0
\(25\) −13.4663 −2.69327
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.46054 −0.456912 −0.228456 0.973554i \(-0.573368\pi\)
−0.228456 + 0.973554i \(0.573368\pi\)
\(30\) 0 0
\(31\) −5.69764 −1.02333 −0.511663 0.859186i \(-0.670970\pi\)
−0.511663 + 0.859186i \(0.670970\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.33530 −0.383921 −0.191961 0.981403i \(-0.561485\pi\)
−0.191961 + 0.981403i \(0.561485\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 5.14822i − 0.804018i −0.915636 0.402009i \(-0.868312\pi\)
0.915636 0.402009i \(-0.131688\pi\)
\(42\) 0 0
\(43\) 13.0199i 1.98552i 0.120117 + 0.992760i \(0.461673\pi\)
−0.120117 + 0.992760i \(0.538327\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −5.35449 −0.781033 −0.390517 0.920596i \(-0.627704\pi\)
−0.390517 + 0.920596i \(0.627704\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.22371 0.580171 0.290085 0.957001i \(-0.406316\pi\)
0.290085 + 0.957001i \(0.406316\pi\)
\(54\) 0 0
\(55\) −21.7553 −2.93349
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −9.60498 −1.25046 −0.625231 0.780440i \(-0.714995\pi\)
−0.625231 + 0.780440i \(0.714995\pi\)
\(60\) 0 0
\(61\) 3.87698i 0.496397i 0.968709 + 0.248198i \(0.0798385\pi\)
−0.968709 + 0.248198i \(0.920162\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 14.5182 1.80076
\(66\) 0 0
\(67\) 4.76342i 0.581944i 0.956732 + 0.290972i \(0.0939787\pi\)
−0.956732 + 0.290972i \(0.906021\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 12.3181i 1.46189i 0.682437 + 0.730944i \(0.260920\pi\)
−0.682437 + 0.730944i \(0.739080\pi\)
\(72\) 0 0
\(73\) − 11.5102i − 1.34716i −0.739113 0.673581i \(-0.764755\pi\)
0.739113 0.673581i \(-0.235245\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) − 13.9166i − 1.56574i −0.622185 0.782870i \(-0.713755\pi\)
0.622185 0.782870i \(-0.286245\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −7.32191 −0.803685 −0.401842 0.915709i \(-0.631630\pi\)
−0.401842 + 0.915709i \(0.631630\pi\)
\(84\) 0 0
\(85\) −11.8884 −1.28948
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 14.0448i − 1.48874i −0.667765 0.744372i \(-0.732749\pi\)
0.667765 0.744372i \(-0.267251\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 20.2108i 2.07358i
\(96\) 0 0
\(97\) 14.5716i 1.47952i 0.672868 + 0.739762i \(0.265062\pi\)
−0.672868 + 0.739762i \(0.734938\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 13.1087i 1.30436i 0.758063 + 0.652181i \(0.226146\pi\)
−0.758063 + 0.652181i \(0.773854\pi\)
\(102\) 0 0
\(103\) 0.209698 0.0206622 0.0103311 0.999947i \(-0.496711\pi\)
0.0103311 + 0.999947i \(0.496711\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.42657i 0.621280i 0.950528 + 0.310640i \(0.100543\pi\)
−0.950528 + 0.310640i \(0.899457\pi\)
\(108\) 0 0
\(109\) 0.940024 0.0900379 0.0450190 0.998986i \(-0.485665\pi\)
0.0450190 + 0.998986i \(0.485665\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1.09821 −0.103311 −0.0516554 0.998665i \(-0.516450\pi\)
−0.0516554 + 0.998665i \(0.516450\pi\)
\(114\) 0 0
\(115\) 20.7609 1.93596
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −14.6301 −1.33001
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 36.3820i − 3.25411i
\(126\) 0 0
\(127\) − 5.22625i − 0.463755i −0.972745 0.231877i \(-0.925513\pi\)
0.972745 0.231877i \(-0.0744868\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −0.875015 −0.0764504 −0.0382252 0.999269i \(-0.512170\pi\)
−0.0382252 + 0.999269i \(0.512170\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −0.236838 −0.0202344 −0.0101172 0.999949i \(-0.503220\pi\)
−0.0101172 + 0.999949i \(0.503220\pi\)
\(138\) 0 0
\(139\) −3.00555 −0.254927 −0.127464 0.991843i \(-0.540684\pi\)
−0.127464 + 0.991843i \(0.540684\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 17.1040 1.43031
\(144\) 0 0
\(145\) − 10.5736i − 0.878087i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 14.8017 1.21260 0.606299 0.795237i \(-0.292653\pi\)
0.606299 + 0.795237i \(0.292653\pi\)
\(150\) 0 0
\(151\) 12.3654i 1.00628i 0.864205 + 0.503140i \(0.167822\pi\)
−0.864205 + 0.503140i \(0.832178\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 24.4842i − 1.96662i
\(156\) 0 0
\(157\) − 10.8906i − 0.869164i −0.900632 0.434582i \(-0.856896\pi\)
0.900632 0.434582i \(-0.143104\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) − 1.51024i − 0.118291i −0.998249 0.0591455i \(-0.981162\pi\)
0.998249 0.0591455i \(-0.0188376\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −10.3333 −0.799612 −0.399806 0.916600i \(-0.630922\pi\)
−0.399806 + 0.916600i \(0.630922\pi\)
\(168\) 0 0
\(169\) 1.58579 0.121984
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 3.35642i − 0.255184i −0.991827 0.127592i \(-0.959275\pi\)
0.991827 0.127592i \(-0.0407248\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 13.2589i 0.991018i 0.868603 + 0.495509i \(0.165018\pi\)
−0.868603 + 0.495509i \(0.834982\pi\)
\(180\) 0 0
\(181\) − 11.8519i − 0.880946i −0.897766 0.440473i \(-0.854811\pi\)
0.897766 0.440473i \(-0.145189\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 10.0354i − 0.737816i
\(186\) 0 0
\(187\) −14.0058 −1.02421
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 1.34660i − 0.0974368i −0.998813 0.0487184i \(-0.984486\pi\)
0.998813 0.0487184i \(-0.0155137\pi\)
\(192\) 0 0
\(193\) −26.1154 −1.87982 −0.939912 0.341416i \(-0.889094\pi\)
−0.939912 + 0.341416i \(0.889094\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 7.49083 0.533699 0.266850 0.963738i \(-0.414017\pi\)
0.266850 + 0.963738i \(0.414017\pi\)
\(198\) 0 0
\(199\) 4.49349 0.318535 0.159267 0.987235i \(-0.449087\pi\)
0.159267 + 0.987235i \(0.449087\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 22.1232 1.54515
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 23.8104i 1.64700i
\(210\) 0 0
\(211\) − 5.93122i − 0.408322i −0.978937 0.204161i \(-0.934553\pi\)
0.978937 0.204161i \(-0.0654466\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −55.9498 −3.81575
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 9.34665 0.628724
\(222\) 0 0
\(223\) 20.6163 1.38057 0.690286 0.723537i \(-0.257485\pi\)
0.690286 + 0.723537i \(0.257485\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −20.4911 −1.36004 −0.680021 0.733193i \(-0.738029\pi\)
−0.680021 + 0.733193i \(0.738029\pi\)
\(228\) 0 0
\(229\) − 29.0409i − 1.91908i −0.281576 0.959539i \(-0.590857\pi\)
0.281576 0.959539i \(-0.409143\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −17.7364 −1.16195 −0.580975 0.813922i \(-0.697328\pi\)
−0.580975 + 0.813922i \(0.697328\pi\)
\(234\) 0 0
\(235\) − 23.0096i − 1.50098i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 20.4248i − 1.32117i −0.750750 0.660586i \(-0.770308\pi\)
0.750750 0.660586i \(-0.229692\pi\)
\(240\) 0 0
\(241\) 3.25742i 0.209829i 0.994481 + 0.104915i \(0.0334569\pi\)
−0.994481 + 0.104915i \(0.966543\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 15.8897i − 1.01104i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −3.95633 −0.249721 −0.124861 0.992174i \(-0.539848\pi\)
−0.124861 + 0.992174i \(0.539848\pi\)
\(252\) 0 0
\(253\) 24.4585 1.53769
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 5.14822i 0.321137i 0.987025 + 0.160569i \(0.0513328\pi\)
−0.987025 + 0.160569i \(0.948667\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0.0785021i 0.00484065i 0.999997 + 0.00242032i \(0.000770414\pi\)
−0.999997 + 0.00242032i \(0.999230\pi\)
\(264\) 0 0
\(265\) 18.1503i 1.11497i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0.883296i 0.0538555i 0.999637 + 0.0269277i \(0.00857240\pi\)
−0.999637 + 0.0269277i \(0.991428\pi\)
\(270\) 0 0
\(271\) −19.3084 −1.17290 −0.586451 0.809984i \(-0.699476\pi\)
−0.586451 + 0.809984i \(0.699476\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 68.1749i − 4.11110i
\(276\) 0 0
\(277\) 16.9327 1.01739 0.508694 0.860948i \(-0.330129\pi\)
0.508694 + 0.860948i \(0.330129\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −5.05417 −0.301507 −0.150753 0.988571i \(-0.548170\pi\)
−0.150753 + 0.988571i \(0.548170\pi\)
\(282\) 0 0
\(283\) −19.1156 −1.13631 −0.568153 0.822923i \(-0.692342\pi\)
−0.568153 + 0.822923i \(0.692342\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 9.34639 0.549788
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 2.37837i − 0.138946i −0.997584 0.0694730i \(-0.977868\pi\)
0.997584 0.0694730i \(-0.0221318\pi\)
\(294\) 0 0
\(295\) − 41.2750i − 2.40312i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −16.3222 −0.943936
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −16.6604 −0.953969
\(306\) 0 0
\(307\) 4.89599 0.279429 0.139714 0.990192i \(-0.455382\pi\)
0.139714 + 0.990192i \(0.455382\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −10.4800 −0.594266 −0.297133 0.954836i \(-0.596030\pi\)
−0.297133 + 0.954836i \(0.596030\pi\)
\(312\) 0 0
\(313\) − 9.60172i − 0.542722i −0.962478 0.271361i \(-0.912526\pi\)
0.962478 0.271361i \(-0.0874737\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 16.7934 0.943214 0.471607 0.881809i \(-0.343674\pi\)
0.471607 + 0.881809i \(0.343674\pi\)
\(318\) 0 0
\(319\) − 12.4568i − 0.697446i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 13.0114i 0.723976i
\(324\) 0 0
\(325\) 45.4960i 2.52366i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 26.8362i 1.47505i 0.675318 + 0.737526i \(0.264006\pi\)
−0.675318 + 0.737526i \(0.735994\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −20.4696 −1.11837
\(336\) 0 0
\(337\) 23.0827 1.25739 0.628697 0.777651i \(-0.283589\pi\)
0.628697 + 0.777651i \(0.283589\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 28.8450i − 1.56204i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1.30690i 0.0701580i 0.999385 + 0.0350790i \(0.0111683\pi\)
−0.999385 + 0.0350790i \(0.988832\pi\)
\(348\) 0 0
\(349\) − 26.5489i − 1.42113i −0.703633 0.710564i \(-0.748440\pi\)
0.703633 0.710564i \(-0.251560\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 16.8893i 0.898928i 0.893298 + 0.449464i \(0.148385\pi\)
−0.893298 + 0.449464i \(0.851615\pi\)
\(354\) 0 0
\(355\) −52.9339 −2.80944
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 33.5901i 1.77282i 0.462904 + 0.886408i \(0.346807\pi\)
−0.462904 + 0.886408i \(0.653193\pi\)
\(360\) 0 0
\(361\) 3.11995 0.164208
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 49.4620 2.58896
\(366\) 0 0
\(367\) −7.35978 −0.384178 −0.192089 0.981378i \(-0.561526\pi\)
−0.192089 + 0.981378i \(0.561526\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 3.63008 0.187958 0.0939791 0.995574i \(-0.470041\pi\)
0.0939791 + 0.995574i \(0.470041\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 8.31293i 0.428138i
\(378\) 0 0
\(379\) − 13.6647i − 0.701908i −0.936393 0.350954i \(-0.885857\pi\)
0.936393 0.350954i \(-0.114143\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 13.6569 0.697833 0.348916 0.937154i \(-0.386550\pi\)
0.348916 + 0.937154i \(0.386550\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −26.9080 −1.36429 −0.682144 0.731218i \(-0.738952\pi\)
−0.682144 + 0.731218i \(0.738952\pi\)
\(390\) 0 0
\(391\) 13.3656 0.675927
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 59.8031 3.00902
\(396\) 0 0
\(397\) 36.7242i 1.84314i 0.388216 + 0.921568i \(0.373091\pi\)
−0.388216 + 0.921568i \(0.626909\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 27.6816 1.38235 0.691176 0.722686i \(-0.257093\pi\)
0.691176 + 0.722686i \(0.257093\pi\)
\(402\) 0 0
\(403\) 19.2494i 0.958883i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 11.8227i − 0.586032i
\(408\) 0 0
\(409\) 8.42820i 0.416747i 0.978049 + 0.208374i \(0.0668170\pi\)
−0.978049 + 0.208374i \(0.933183\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) − 31.4641i − 1.54451i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 18.3029 0.894154 0.447077 0.894495i \(-0.352465\pi\)
0.447077 + 0.894495i \(0.352465\pi\)
\(420\) 0 0
\(421\) 22.6274 1.10279 0.551396 0.834243i \(-0.314095\pi\)
0.551396 + 0.834243i \(0.314095\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 37.2549i − 1.80713i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 4.44906i − 0.214304i −0.994243 0.107152i \(-0.965827\pi\)
0.994243 0.107152i \(-0.0341731\pi\)
\(432\) 0 0
\(433\) 1.66205i 0.0798730i 0.999202 + 0.0399365i \(0.0127156\pi\)
−0.999202 + 0.0399365i \(0.987284\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 22.7220i − 1.08694i
\(438\) 0 0
\(439\) 18.7551 0.895130 0.447565 0.894251i \(-0.352291\pi\)
0.447565 + 0.894251i \(0.352291\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 33.0314i − 1.56937i −0.619896 0.784684i \(-0.712825\pi\)
0.619896 0.784684i \(-0.287175\pi\)
\(444\) 0 0
\(445\) 60.3539 2.86105
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −9.29441 −0.438630 −0.219315 0.975654i \(-0.570382\pi\)
−0.219315 + 0.975654i \(0.570382\pi\)
\(450\) 0 0
\(451\) 26.0635 1.22728
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −27.4448 −1.28381 −0.641906 0.766784i \(-0.721856\pi\)
−0.641906 + 0.766784i \(0.721856\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 4.28209i − 0.199437i −0.995016 0.0997183i \(-0.968206\pi\)
0.995016 0.0997183i \(-0.0317942\pi\)
\(462\) 0 0
\(463\) 16.1278i 0.749521i 0.927122 + 0.374760i \(0.122275\pi\)
−0.927122 + 0.374760i \(0.877725\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −27.5403 −1.27441 −0.637207 0.770692i \(-0.719910\pi\)
−0.637207 + 0.770692i \(0.719910\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −65.9149 −3.03077
\(474\) 0 0
\(475\) −63.3347 −2.90600
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 19.5509 0.893304 0.446652 0.894708i \(-0.352616\pi\)
0.446652 + 0.894708i \(0.352616\pi\)
\(480\) 0 0
\(481\) 7.88980i 0.359744i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −62.6179 −2.84333
\(486\) 0 0
\(487\) 37.7917i 1.71250i 0.516558 + 0.856252i \(0.327213\pi\)
−0.516558 + 0.856252i \(0.672787\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 29.2605i − 1.32051i −0.751042 0.660254i \(-0.770449\pi\)
0.751042 0.660254i \(-0.229551\pi\)
\(492\) 0 0
\(493\) − 6.80713i − 0.306578i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 12.6422i 0.565944i 0.959128 + 0.282972i \(0.0913203\pi\)
−0.959128 + 0.282972i \(0.908680\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −27.0714 −1.20706 −0.603528 0.797342i \(-0.706239\pi\)
−0.603528 + 0.797342i \(0.706239\pi\)
\(504\) 0 0
\(505\) −56.3312 −2.50671
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 11.6983i − 0.518520i −0.965808 0.259260i \(-0.916521\pi\)
0.965808 0.259260i \(-0.0834786\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0.901124i 0.0397083i
\(516\) 0 0
\(517\) − 27.1077i − 1.19220i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 7.20643i 0.315719i 0.987462 + 0.157860i \(0.0504594\pi\)
−0.987462 + 0.157860i \(0.949541\pi\)
\(522\) 0 0
\(523\) −17.7145 −0.774602 −0.387301 0.921953i \(-0.626593\pi\)
−0.387301 + 0.921953i \(0.626593\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 15.7626i − 0.686630i
\(528\) 0 0
\(529\) −0.340485 −0.0148037
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −17.3932 −0.753385
\(534\) 0 0
\(535\) −27.6166 −1.19397
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 23.1180 0.993921 0.496961 0.867773i \(-0.334449\pi\)
0.496961 + 0.867773i \(0.334449\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 4.03951i 0.173034i
\(546\) 0 0
\(547\) − 16.0524i − 0.686351i −0.939271 0.343176i \(-0.888497\pi\)
0.939271 0.343176i \(-0.111503\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −11.5724 −0.493001
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 12.5808 0.533067 0.266533 0.963826i \(-0.414122\pi\)
0.266533 + 0.963826i \(0.414122\pi\)
\(558\) 0 0
\(559\) 43.9877 1.86048
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −36.5948 −1.54229 −0.771144 0.636660i \(-0.780315\pi\)
−0.771144 + 0.636660i \(0.780315\pi\)
\(564\) 0 0
\(565\) − 4.71927i − 0.198541i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 24.4989 1.02705 0.513524 0.858075i \(-0.328340\pi\)
0.513524 + 0.858075i \(0.328340\pi\)
\(570\) 0 0
\(571\) 32.7924i 1.37232i 0.727451 + 0.686159i \(0.240705\pi\)
−0.727451 + 0.686159i \(0.759295\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 65.0586i 2.71313i
\(576\) 0 0
\(577\) − 19.2140i − 0.799888i −0.916540 0.399944i \(-0.869030\pi\)
0.916540 0.399944i \(-0.130970\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 21.3830i 0.885594i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −24.1451 −0.996573 −0.498286 0.867012i \(-0.666037\pi\)
−0.498286 + 0.867012i \(0.666037\pi\)
\(588\) 0 0
\(589\) −26.7971 −1.10415
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 2.54582i − 0.104544i −0.998633 0.0522722i \(-0.983354\pi\)
0.998633 0.0522722i \(-0.0166463\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 14.9899i − 0.612469i −0.951956 0.306234i \(-0.900931\pi\)
0.951956 0.306234i \(-0.0990691\pi\)
\(600\) 0 0
\(601\) − 24.3343i − 0.992618i −0.868146 0.496309i \(-0.834688\pi\)
0.868146 0.496309i \(-0.165312\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 62.8691i − 2.55599i
\(606\) 0 0
\(607\) 8.91288 0.361763 0.180881 0.983505i \(-0.442105\pi\)
0.180881 + 0.983505i \(0.442105\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 18.0901i 0.731848i
\(612\) 0 0
\(613\) −17.1748 −0.693685 −0.346842 0.937923i \(-0.612746\pi\)
−0.346842 + 0.937923i \(0.612746\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −35.9980 −1.44922 −0.724612 0.689157i \(-0.757981\pi\)
−0.724612 + 0.689157i \(0.757981\pi\)
\(618\) 0 0
\(619\) −4.11123 −0.165244 −0.0826222 0.996581i \(-0.526329\pi\)
−0.0826222 + 0.996581i \(0.526329\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 89.0108 3.56043
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 6.46065i − 0.257603i
\(630\) 0 0
\(631\) 34.8970i 1.38923i 0.719383 + 0.694613i \(0.244425\pi\)
−0.719383 + 0.694613i \(0.755575\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 22.4585 0.891239
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −7.32735 −0.289413 −0.144707 0.989475i \(-0.546224\pi\)
−0.144707 + 0.989475i \(0.546224\pi\)
\(642\) 0 0
\(643\) 9.24275 0.364498 0.182249 0.983252i \(-0.441662\pi\)
0.182249 + 0.983252i \(0.441662\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 15.1803 0.596798 0.298399 0.954441i \(-0.403547\pi\)
0.298399 + 0.954441i \(0.403547\pi\)
\(648\) 0 0
\(649\) − 48.6263i − 1.90875i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 2.12849 0.0832943 0.0416471 0.999132i \(-0.486739\pi\)
0.0416471 + 0.999132i \(0.486739\pi\)
\(654\) 0 0
\(655\) − 3.76016i − 0.146922i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 13.0520i 0.508436i 0.967147 + 0.254218i \(0.0818180\pi\)
−0.967147 + 0.254218i \(0.918182\pi\)
\(660\) 0 0
\(661\) − 19.2786i − 0.749851i −0.927055 0.374926i \(-0.877668\pi\)
0.927055 0.374926i \(-0.122332\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 11.8874i 0.460281i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −19.6277 −0.757718
\(672\) 0 0
\(673\) −7.08216 −0.272997 −0.136499 0.990640i \(-0.543585\pi\)
−0.136499 + 0.990640i \(0.543585\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 36.5042i 1.40297i 0.712685 + 0.701485i \(0.247479\pi\)
−0.712685 + 0.701485i \(0.752521\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 30.1206i 1.15253i 0.817261 + 0.576267i \(0.195491\pi\)
−0.817261 + 0.576267i \(0.804509\pi\)
\(684\) 0 0
\(685\) − 1.01775i − 0.0388863i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 14.2698i − 0.543635i
\(690\) 0 0
\(691\) 39.8113 1.51449 0.757247 0.653129i \(-0.226544\pi\)
0.757247 + 0.653129i \(0.226544\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 12.9156i − 0.489916i
\(696\) 0 0
\(697\) 14.2426 0.539478
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −28.0795 −1.06055 −0.530275 0.847826i \(-0.677911\pi\)
−0.530275 + 0.847826i \(0.677911\pi\)
\(702\) 0 0
\(703\) −10.9834 −0.414245
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −36.9475 −1.38759 −0.693797 0.720170i \(-0.744064\pi\)
−0.693797 + 0.720170i \(0.744064\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 27.5264i 1.03087i
\(714\) 0 0
\(715\) 73.5002i 2.74875i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −18.7090 −0.697728 −0.348864 0.937173i \(-0.613432\pi\)
−0.348864 + 0.937173i \(0.613432\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 33.1345 1.23059
\(726\) 0 0
\(727\) −31.2078 −1.15743 −0.578716 0.815529i \(-0.696446\pi\)
−0.578716 + 0.815529i \(0.696446\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −36.0198 −1.33224
\(732\) 0 0
\(733\) 4.06897i 0.150291i 0.997173 + 0.0751455i \(0.0239421\pi\)
−0.997173 + 0.0751455i \(0.976058\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −24.1154 −0.888301
\(738\) 0 0
\(739\) 39.9025i 1.46784i 0.679237 + 0.733919i \(0.262311\pi\)
−0.679237 + 0.733919i \(0.737689\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 41.7530i − 1.53177i −0.642979 0.765884i \(-0.722302\pi\)
0.642979 0.765884i \(-0.277698\pi\)
\(744\) 0 0
\(745\) 63.6064i 2.33036i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) − 9.09927i − 0.332037i −0.986123 0.166019i \(-0.946909\pi\)
0.986123 0.166019i \(-0.0530911\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −53.1371 −1.93386
\(756\) 0 0
\(757\) −0.902155 −0.0327894 −0.0163947 0.999866i \(-0.505219\pi\)
−0.0163947 + 0.999866i \(0.505219\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 23.7781i 0.861955i 0.902363 + 0.430978i \(0.141831\pi\)
−0.902363 + 0.430978i \(0.858169\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 32.4503i 1.17171i
\(768\) 0 0
\(769\) − 16.3028i − 0.587895i −0.955822 0.293948i \(-0.905031\pi\)
0.955822 0.293948i \(-0.0949691\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 25.0881i 0.902358i 0.892434 + 0.451179i \(0.148996\pi\)
−0.892434 + 0.451179i \(0.851004\pi\)
\(774\) 0 0
\(775\) 76.7264 2.75609
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 24.2131i − 0.867523i
\(780\) 0 0
\(781\) −62.3618 −2.23148
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 46.7996 1.67035
\(786\) 0 0
\(787\) −4.47130 −0.159385 −0.0796924 0.996820i \(-0.525394\pi\)
−0.0796924 + 0.996820i \(0.525394\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 13.0984 0.465136
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 6.62320i 0.234606i 0.993096 + 0.117303i \(0.0374248\pi\)
−0.993096 + 0.117303i \(0.962575\pi\)
\(798\) 0 0
\(799\) − 14.8133i − 0.524056i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 58.2715 2.05636
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 8.37044 0.294289 0.147144 0.989115i \(-0.452992\pi\)
0.147144 + 0.989115i \(0.452992\pi\)
\(810\) 0 0
\(811\) −1.20944 −0.0424692 −0.0212346 0.999775i \(-0.506760\pi\)
−0.0212346 + 0.999775i \(0.506760\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 6.48988 0.227330
\(816\) 0 0
\(817\) 61.2351i 2.14235i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −51.2519 −1.78871 −0.894353 0.447363i \(-0.852363\pi\)
−0.894353 + 0.447363i \(0.852363\pi\)
\(822\) 0 0
\(823\) − 31.5546i − 1.09992i −0.835189 0.549962i \(-0.814642\pi\)
0.835189 0.549962i \(-0.185358\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 23.4800i − 0.816480i −0.912875 0.408240i \(-0.866143\pi\)
0.912875 0.408240i \(-0.133857\pi\)
\(828\) 0 0
\(829\) − 17.2639i − 0.599599i −0.954002 0.299800i \(-0.903080\pi\)
0.954002 0.299800i \(-0.0969199\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) − 44.4046i − 1.53668i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 37.4781 1.29389 0.646943 0.762538i \(-0.276047\pi\)
0.646943 + 0.762538i \(0.276047\pi\)
\(840\) 0 0
\(841\) −22.9457 −0.791232
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 6.81452i 0.234427i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 11.2823i 0.386753i
\(852\) 0 0
\(853\) − 9.50115i − 0.325313i −0.986683 0.162657i \(-0.947994\pi\)
0.986683 0.162657i \(-0.0520063\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 32.7797i 1.11973i 0.828583 + 0.559866i \(0.189147\pi\)
−0.828583 + 0.559866i \(0.810853\pi\)
\(858\) 0 0
\(859\) 5.37306 0.183327 0.0916633 0.995790i \(-0.470782\pi\)
0.0916633 + 0.995790i \(0.470782\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 29.6824i 1.01040i 0.863003 + 0.505200i \(0.168581\pi\)
−0.863003 + 0.505200i \(0.831419\pi\)
\(864\) 0 0
\(865\) 14.4234 0.490409
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 70.4544 2.39000
\(870\) 0 0
\(871\) 16.0932 0.545296
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −30.3242 −1.02398 −0.511988 0.858993i \(-0.671091\pi\)
−0.511988 + 0.858993i \(0.671091\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 16.5012i 0.555939i 0.960590 + 0.277969i \(0.0896614\pi\)
−0.960590 + 0.277969i \(0.910339\pi\)
\(882\) 0 0
\(883\) 22.5909i 0.760245i 0.924936 + 0.380122i \(0.124118\pi\)
−0.924936 + 0.380122i \(0.875882\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 38.6036 1.29618 0.648090 0.761563i \(-0.275568\pi\)
0.648090 + 0.761563i \(0.275568\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −25.1832 −0.842723
\(894\) 0 0
\(895\) −56.9769 −1.90453
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 14.0193 0.467570
\(900\) 0 0
\(901\) 11.6849i 0.389282i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 50.9307 1.69299
\(906\) 0 0
\(907\) 36.6743i 1.21775i 0.793266 + 0.608875i \(0.208379\pi\)
−0.793266 + 0.608875i \(0.791621\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 52.6835i 1.74548i 0.488185 + 0.872740i \(0.337659\pi\)
−0.488185 + 0.872740i \(0.662341\pi\)
\(912\) 0 0
\(913\) − 37.0680i − 1.22677i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 5.11618i 0.168767i 0.996433 + 0.0843836i \(0.0268921\pi\)
−0.996433 + 0.0843836i \(0.973108\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 41.6166 1.36983
\(924\) 0 0
\(925\) 31.4480 1.03400
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 35.3694i 1.16043i 0.814462 + 0.580217i \(0.197032\pi\)
−0.814462 + 0.580217i \(0.802968\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) − 60.1864i − 1.96831i
\(936\) 0 0
\(937\) 23.6290i 0.771924i 0.922515 + 0.385962i \(0.126130\pi\)
−0.922515 + 0.385962i \(0.873870\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 19.2187i − 0.626510i −0.949669 0.313255i \(-0.898581\pi\)
0.949669 0.313255i \(-0.101419\pi\)
\(942\) 0 0
\(943\) −24.8721 −0.809947
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 27.8159i 0.903894i 0.892045 + 0.451947i \(0.149270\pi\)
−0.892045 + 0.451947i \(0.850730\pi\)
\(948\) 0 0
\(949\) −38.8870 −1.26232
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −56.8087 −1.84021 −0.920107 0.391668i \(-0.871898\pi\)
−0.920107 + 0.391668i \(0.871898\pi\)
\(954\) 0 0
\(955\) 5.78669 0.187253
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 1.46310 0.0471967
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 112.224i − 3.61262i
\(966\) 0 0
\(967\) 20.3860i 0.655570i 0.944752 + 0.327785i \(0.106302\pi\)
−0.944752 + 0.327785i \(0.893698\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −11.2287 −0.360347 −0.180174 0.983635i \(-0.557666\pi\)
−0.180174 + 0.983635i \(0.557666\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 12.1833 0.389778 0.194889 0.980825i \(-0.437565\pi\)
0.194889 + 0.980825i \(0.437565\pi\)
\(978\) 0 0
\(979\) 71.1033 2.27247
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 9.06374 0.289088 0.144544 0.989498i \(-0.453828\pi\)
0.144544 + 0.989498i \(0.453828\pi\)
\(984\) 0 0
\(985\) 32.1899i 1.02566i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 62.9018 2.00016
\(990\) 0 0
\(991\) − 5.37158i − 0.170634i −0.996354 0.0853170i \(-0.972810\pi\)
0.996354 0.0853170i \(-0.0271903\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 19.3096i 0.612157i
\(996\) 0 0
\(997\) 14.3440i 0.454278i 0.973862 + 0.227139i \(0.0729372\pi\)
−0.973862 + 0.227139i \(0.927063\pi\)
\(998\) 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.w.1567.8 8
3.2 odd 2 2352.2.b.l.1567.1 yes 8
4.3 odd 2 7056.2.b.x.1567.8 8
7.6 odd 2 7056.2.b.x.1567.1 8
12.11 even 2 2352.2.b.k.1567.1 8
21.2 odd 6 2352.2.bl.q.31.1 8
21.5 even 6 2352.2.bl.t.31.4 8
21.11 odd 6 2352.2.bl.o.607.4 8
21.17 even 6 2352.2.bl.r.607.1 8
21.20 even 2 2352.2.b.k.1567.8 yes 8
28.27 even 2 inner 7056.2.b.w.1567.1 8
84.11 even 6 2352.2.bl.t.607.4 8
84.23 even 6 2352.2.bl.r.31.1 8
84.47 odd 6 2352.2.bl.o.31.4 8
84.59 odd 6 2352.2.bl.q.607.1 8
84.83 odd 2 2352.2.b.l.1567.8 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2352.2.b.k.1567.1 8 12.11 even 2
2352.2.b.k.1567.8 yes 8 21.20 even 2
2352.2.b.l.1567.1 yes 8 3.2 odd 2
2352.2.b.l.1567.8 yes 8 84.83 odd 2
2352.2.bl.o.31.4 8 84.47 odd 6
2352.2.bl.o.607.4 8 21.11 odd 6
2352.2.bl.q.31.1 8 21.2 odd 6
2352.2.bl.q.607.1 8 84.59 odd 6
2352.2.bl.r.31.1 8 84.23 even 6
2352.2.bl.r.607.1 8 21.17 even 6
2352.2.bl.t.31.4 8 21.5 even 6
2352.2.bl.t.607.4 8 84.11 even 6
7056.2.b.w.1567.1 8 28.27 even 2 inner
7056.2.b.w.1567.8 8 1.1 even 1 trivial
7056.2.b.x.1567.1 8 7.6 odd 2
7056.2.b.x.1567.8 8 4.3 odd 2