Properties

Label 7056.2.k.g.881.6
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 881.6
Root \(0.923880 + 0.382683i\) of defining polynomial
Character \(\chi\) \(=\) 7056.881
Dual form 7056.2.k.g.881.5

$q$-expansion

\(f(q)\) \(=\) \(q+2.93015 q^{5} +O(q^{10})\) \(q+2.93015 q^{5} +4.82843i q^{11} +2.93015i q^{13} -7.07401 q^{17} -5.86030i q^{19} -2.00000i q^{23} +3.58579 q^{25} +0.828427i q^{29} +5.86030i q^{31} -5.41421 q^{37} -1.21371 q^{41} -4.48528 q^{43} -5.86030 q^{47} +7.07107i q^{53} +14.1480i q^{55} -5.86030 q^{59} +1.21371i q^{61} +8.58579i q^{65} +8.48528 q^{67} +0.828427i q^{71} +7.07401i q^{73} -1.65685 q^{79} +11.7206 q^{83} -20.7279 q^{85} -11.2179 q^{89} -17.1716i q^{95} -7.07401i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + O(q^{10}) \) \( 8 q + 40 q^{25} - 32 q^{37} + 32 q^{43} + 32 q^{79} - 64 q^{85} + O(q^{100}) \)

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
\(5\) 2.93015 1.31040 0.655202 0.755454i \(-0.272584\pi\)
0.655202 + 0.755454i \(0.272584\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.82843i 1.45583i 0.685670 + 0.727913i \(0.259509\pi\)
−0.685670 + 0.727913i \(0.740491\pi\)
\(12\) 0 0
\(13\) 2.93015i 0.812678i 0.913722 + 0.406339i \(0.133195\pi\)
−0.913722 + 0.406339i \(0.866805\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −7.07401 −1.71570 −0.857850 0.513900i \(-0.828200\pi\)
−0.857850 + 0.513900i \(0.828200\pi\)
\(18\) 0 0
\(19\) − 5.86030i − 1.34445i −0.740349 0.672223i \(-0.765340\pi\)
0.740349 0.672223i \(-0.234660\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 2.00000i − 0.417029i −0.978019 0.208514i \(-0.933137\pi\)
0.978019 0.208514i \(-0.0668628\pi\)
\(24\) 0 0
\(25\) 3.58579 0.717157
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.828427i 0.153835i 0.997037 + 0.0769175i \(0.0245078\pi\)
−0.997037 + 0.0769175i \(0.975492\pi\)
\(30\) 0 0
\(31\) 5.86030i 1.05254i 0.850317 + 0.526271i \(0.176410\pi\)
−0.850317 + 0.526271i \(0.823590\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.41421 −0.890091 −0.445046 0.895508i \(-0.646813\pi\)
−0.445046 + 0.895508i \(0.646813\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.21371 −0.189549 −0.0947747 0.995499i \(-0.530213\pi\)
−0.0947747 + 0.995499i \(0.530213\pi\)
\(42\) 0 0
\(43\) −4.48528 −0.683999 −0.341999 0.939700i \(-0.611104\pi\)
−0.341999 + 0.939700i \(0.611104\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −5.86030 −0.854813 −0.427406 0.904060i \(-0.640573\pi\)
−0.427406 + 0.904060i \(0.640573\pi\)
\(48\) 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
\(55\) 14.1480i 1.90772i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −5.86030 −0.762946 −0.381473 0.924380i \(-0.624583\pi\)
−0.381473 + 0.924380i \(0.624583\pi\)
\(60\) 0 0
\(61\) 1.21371i 0.155399i 0.996977 + 0.0776997i \(0.0247575\pi\)
−0.996977 + 0.0776997i \(0.975242\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 8.58579i 1.06494i
\(66\) 0 0
\(67\) 8.48528 1.03664 0.518321 0.855186i \(-0.326557\pi\)
0.518321 + 0.855186i \(0.326557\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0.828427i 0.0983162i 0.998791 + 0.0491581i \(0.0156538\pi\)
−0.998791 + 0.0491581i \(0.984346\pi\)
\(72\) 0 0
\(73\) 7.07401i 0.827950i 0.910288 + 0.413975i \(0.135860\pi\)
−0.910288 + 0.413975i \(0.864140\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −1.65685 −0.186411 −0.0932053 0.995647i \(-0.529711\pi\)
−0.0932053 + 0.995647i \(0.529711\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 11.7206 1.28650 0.643252 0.765655i \(-0.277585\pi\)
0.643252 + 0.765655i \(0.277585\pi\)
\(84\) 0 0
\(85\) −20.7279 −2.24826
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −11.2179 −1.18909 −0.594546 0.804062i \(-0.702668\pi\)
−0.594546 + 0.804062i \(0.702668\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 17.1716i − 1.76177i
\(96\) 0 0
\(97\) − 7.07401i − 0.718257i −0.933288 0.359128i \(-0.883074\pi\)
0.933288 0.359128i \(-0.116926\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −5.35757 −0.533098 −0.266549 0.963821i \(-0.585883\pi\)
−0.266549 + 0.963821i \(0.585883\pi\)
\(102\) 0 0
\(103\) 14.1480i 1.39405i 0.717049 + 0.697023i \(0.245492\pi\)
−0.717049 + 0.697023i \(0.754508\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 13.3137i − 1.28708i −0.765410 0.643542i \(-0.777464\pi\)
0.765410 0.643542i \(-0.222536\pi\)
\(108\) 0 0
\(109\) −5.41421 −0.518588 −0.259294 0.965798i \(-0.583490\pi\)
−0.259294 + 0.965798i \(0.583490\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 13.8995i 1.30755i 0.756687 + 0.653777i \(0.226817\pi\)
−0.756687 + 0.653777i \(0.773183\pi\)
\(114\) 0 0
\(115\) − 5.86030i − 0.546476i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −12.3137 −1.11943
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −4.14386 −0.370638
\(126\) 0 0
\(127\) −16.4853 −1.46283 −0.731416 0.681931i \(-0.761140\pi\)
−0.731416 + 0.681931i \(0.761140\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −8.28772 −0.724101 −0.362051 0.932158i \(-0.617923\pi\)
−0.362051 + 0.932158i \(0.617923\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.82843i 0.412520i 0.978497 + 0.206260i \(0.0661293\pi\)
−0.978497 + 0.206260i \(0.933871\pi\)
\(138\) 0 0
\(139\) − 8.28772i − 0.702955i −0.936196 0.351478i \(-0.885679\pi\)
0.936196 0.351478i \(-0.114321\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −14.1480 −1.18312
\(144\) 0 0
\(145\) 2.42742i 0.201586i
\(146\) 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
\(151\) −4.48528 −0.365007 −0.182504 0.983205i \(-0.558420\pi\)
−0.182504 + 0.983205i \(0.558420\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 17.1716i 1.37925i
\(156\) 0 0
\(157\) − 22.9385i − 1.83069i −0.402671 0.915345i \(-0.631918\pi\)
0.402671 0.915345i \(-0.368082\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −9.17157 −0.718373 −0.359187 0.933266i \(-0.616946\pi\)
−0.359187 + 0.933266i \(0.616946\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 14.1480 1.09481 0.547403 0.836869i \(-0.315616\pi\)
0.547403 + 0.836869i \(0.315616\pi\)
\(168\) 0 0
\(169\) 4.41421 0.339555
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −1.21371 −0.0922765 −0.0461383 0.998935i \(-0.514691\pi\)
−0.0461383 + 0.998935i \(0.514691\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 26.4853i 1.97960i 0.142454 + 0.989801i \(0.454501\pi\)
−0.142454 + 0.989801i \(0.545499\pi\)
\(180\) 0 0
\(181\) 9.50143i 0.706236i 0.935579 + 0.353118i \(0.114879\pi\)
−0.935579 + 0.353118i \(0.885121\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −15.8645 −1.16638
\(186\) 0 0
\(187\) − 34.1563i − 2.49776i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 15.1716i 1.09778i 0.835896 + 0.548888i \(0.184949\pi\)
−0.835896 + 0.548888i \(0.815051\pi\)
\(192\) 0 0
\(193\) −6.34315 −0.456590 −0.228295 0.973592i \(-0.573315\pi\)
−0.228295 + 0.973592i \(0.573315\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 7.75736i 0.552689i 0.961059 + 0.276344i \(0.0891231\pi\)
−0.961059 + 0.276344i \(0.910877\pi\)
\(198\) 0 0
\(199\) − 16.5754i − 1.17500i −0.809224 0.587501i \(-0.800112\pi\)
0.809224 0.587501i \(-0.199888\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −3.55635 −0.248386
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 28.2960 1.95728
\(210\) 0 0
\(211\) 15.3137 1.05424 0.527120 0.849791i \(-0.323272\pi\)
0.527120 + 0.849791i \(0.323272\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −13.1426 −0.896315
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 20.7279i − 1.39431i
\(222\) 0 0
\(223\) − 3.43289i − 0.229883i −0.993372 0.114942i \(-0.963332\pi\)
0.993372 0.114942i \(-0.0366681\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 14.1480 0.939037 0.469519 0.882923i \(-0.344427\pi\)
0.469519 + 0.882923i \(0.344427\pi\)
\(228\) 0 0
\(229\) − 18.7946i − 1.24198i −0.783817 0.620992i \(-0.786730\pi\)
0.783817 0.620992i \(-0.213270\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 5.17157i − 0.338801i −0.985547 0.169401i \(-0.945817\pi\)
0.985547 0.169401i \(-0.0541831\pi\)
\(234\) 0 0
\(235\) −17.1716 −1.12015
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 11.6569i 0.754019i 0.926209 + 0.377010i \(0.123048\pi\)
−0.926209 + 0.377010i \(0.876952\pi\)
\(240\) 0 0
\(241\) − 25.3659i − 1.63396i −0.576665 0.816980i \(-0.695646\pi\)
0.576665 0.816980i \(-0.304354\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 17.1716 1.09260
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −10.7151 −0.676333 −0.338167 0.941086i \(-0.609807\pi\)
−0.338167 + 0.941086i \(0.609807\pi\)
\(252\) 0 0
\(253\) 9.65685 0.607121
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 10.5069 0.655402 0.327701 0.944781i \(-0.393726\pi\)
0.327701 + 0.944781i \(0.393726\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 4.14214i 0.255415i 0.991812 + 0.127708i \(0.0407619\pi\)
−0.991812 + 0.127708i \(0.959238\pi\)
\(264\) 0 0
\(265\) 20.7193i 1.27278i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −2.21918 −0.135306 −0.0676528 0.997709i \(-0.521551\pi\)
−0.0676528 + 0.997709i \(0.521551\pi\)
\(270\) 0 0
\(271\) 5.86030i 0.355988i 0.984032 + 0.177994i \(0.0569608\pi\)
−0.984032 + 0.177994i \(0.943039\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 17.3137i 1.04406i
\(276\) 0 0
\(277\) 12.9706 0.779326 0.389663 0.920958i \(-0.372592\pi\)
0.389663 + 0.920958i \(0.372592\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 13.1716i 0.785750i 0.919592 + 0.392875i \(0.128520\pi\)
−0.919592 + 0.392875i \(0.871480\pi\)
\(282\) 0 0
\(283\) − 9.29319i − 0.552423i −0.961097 0.276211i \(-0.910921\pi\)
0.961097 0.276211i \(-0.0890790\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 33.0416 1.94363
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 22.9385 1.34008 0.670040 0.742325i \(-0.266277\pi\)
0.670040 + 0.742325i \(0.266277\pi\)
\(294\) 0 0
\(295\) −17.1716 −0.999768
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 5.86030 0.338910
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 3.55635i 0.203636i
\(306\) 0 0
\(307\) 22.4357i 1.28048i 0.768177 + 0.640238i \(0.221164\pi\)
−0.768177 + 0.640238i \(0.778836\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −25.8686 −1.46688 −0.733438 0.679757i \(-0.762085\pi\)
−0.733438 + 0.679757i \(0.762085\pi\)
\(312\) 0 0
\(313\) − 5.35757i − 0.302828i −0.988470 0.151414i \(-0.951617\pi\)
0.988470 0.151414i \(-0.0483826\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 32.2426i 1.81093i 0.424424 + 0.905464i \(0.360477\pi\)
−0.424424 + 0.905464i \(0.639523\pi\)
\(318\) 0 0
\(319\) −4.00000 −0.223957
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 41.4558i 2.30666i
\(324\) 0 0
\(325\) 10.5069i 0.582818i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −21.6569 −1.19037 −0.595184 0.803589i \(-0.702921\pi\)
−0.595184 + 0.803589i \(0.702921\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 24.8632 1.35842
\(336\) 0 0
\(337\) −27.0711 −1.47466 −0.737328 0.675535i \(-0.763913\pi\)
−0.737328 + 0.675535i \(0.763913\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −28.2960 −1.53232
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 14.9706i − 0.803662i −0.915714 0.401831i \(-0.868374\pi\)
0.915714 0.401831i \(-0.131626\pi\)
\(348\) 0 0
\(349\) 11.2179i 0.600479i 0.953864 + 0.300239i \(0.0970666\pi\)
−0.953864 + 0.300239i \(0.902933\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 20.5111 1.09169 0.545847 0.837885i \(-0.316208\pi\)
0.545847 + 0.837885i \(0.316208\pi\)
\(354\) 0 0
\(355\) 2.42742i 0.128834i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 34.9706i 1.84568i 0.385189 + 0.922838i \(0.374136\pi\)
−0.385189 + 0.922838i \(0.625864\pi\)
\(360\) 0 0
\(361\) −15.3431 −0.807534
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 20.7279i 1.08495i
\(366\) 0 0
\(367\) − 20.0083i − 1.04443i −0.852815 0.522213i \(-0.825107\pi\)
0.852815 0.522213i \(-0.174893\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 22.6274 1.17160 0.585802 0.810454i \(-0.300780\pi\)
0.585802 + 0.810454i \(0.300780\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −2.42742 −0.125018
\(378\) 0 0
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 23.4412 1.19779 0.598895 0.800828i \(-0.295607\pi\)
0.598895 + 0.800828i \(0.295607\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 25.7990i 1.30806i 0.756468 + 0.654030i \(0.226923\pi\)
−0.756468 + 0.654030i \(0.773077\pi\)
\(390\) 0 0
\(391\) 14.1480i 0.715496i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −4.85483 −0.244273
\(396\) 0 0
\(397\) − 17.7891i − 0.892812i −0.894831 0.446406i \(-0.852704\pi\)
0.894831 0.446406i \(-0.147296\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 5.17157i 0.258256i 0.991628 + 0.129128i \(0.0412178\pi\)
−0.991628 + 0.129128i \(0.958782\pi\)
\(402\) 0 0
\(403\) −17.1716 −0.855377
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 26.1421i − 1.29582i
\(408\) 0 0
\(409\) 9.50143i 0.469815i 0.972018 + 0.234908i \(0.0754788\pi\)
−0.972018 + 0.234908i \(0.924521\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 34.3431 1.68584
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −19.0029 −0.928350 −0.464175 0.885743i \(-0.653649\pi\)
−0.464175 + 0.885743i \(0.653649\pi\)
\(420\) 0 0
\(421\) −0.686292 −0.0334478 −0.0167239 0.999860i \(-0.505324\pi\)
−0.0167239 + 0.999860i \(0.505324\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −25.3659 −1.23043
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 35.4558i − 1.70785i −0.520398 0.853924i \(-0.674216\pi\)
0.520398 0.853924i \(-0.325784\pi\)
\(432\) 0 0
\(433\) − 1.21371i − 0.0583271i −0.999575 0.0291636i \(-0.990716\pi\)
0.999575 0.0291636i \(-0.00928436\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −11.7206 −0.560673
\(438\) 0 0
\(439\) 24.8632i 1.18665i 0.804962 + 0.593327i \(0.202186\pi\)
−0.804962 + 0.593327i \(0.797814\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 19.1716i 0.910869i 0.890269 + 0.455434i \(0.150516\pi\)
−0.890269 + 0.455434i \(0.849484\pi\)
\(444\) 0 0
\(445\) −32.8701 −1.55819
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 15.7574i 0.743636i 0.928306 + 0.371818i \(0.121265\pi\)
−0.928306 + 0.371818i \(0.878735\pi\)
\(450\) 0 0
\(451\) − 5.86030i − 0.275951i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −16.6274 −0.777798 −0.388899 0.921280i \(-0.627144\pi\)
−0.388899 + 0.921280i \(0.627144\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 10.5069 0.489355 0.244677 0.969605i \(-0.421318\pi\)
0.244677 + 0.969605i \(0.421318\pi\)
\(462\) 0 0
\(463\) −20.4853 −0.952032 −0.476016 0.879437i \(-0.657920\pi\)
−0.476016 + 0.879437i \(0.657920\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −14.1480 −0.654692 −0.327346 0.944904i \(-0.606154\pi\)
−0.327346 + 0.944904i \(0.606154\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 21.6569i − 0.995783i
\(474\) 0 0
\(475\) − 21.0138i − 0.964179i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −42.4441 −1.93932 −0.969659 0.244460i \(-0.921389\pi\)
−0.969659 + 0.244460i \(0.921389\pi\)
\(480\) 0 0
\(481\) − 15.8645i − 0.723357i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 20.7279i − 0.941206i
\(486\) 0 0
\(487\) 41.4558 1.87854 0.939272 0.343174i \(-0.111502\pi\)
0.939272 + 0.343174i \(0.111502\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 25.3137i 1.14239i 0.820814 + 0.571196i \(0.193520\pi\)
−0.820814 + 0.571196i \(0.806480\pi\)
\(492\) 0 0
\(493\) − 5.86030i − 0.263935i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −26.1421 −1.17028 −0.585141 0.810931i \(-0.698961\pi\)
−0.585141 + 0.810931i \(0.698961\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 4.85483 0.216466 0.108233 0.994126i \(-0.465481\pi\)
0.108233 + 0.994126i \(0.465481\pi\)
\(504\) 0 0
\(505\) −15.6985 −0.698573
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −26.0769 −1.15584 −0.577918 0.816095i \(-0.696135\pi\)
−0.577918 + 0.816095i \(0.696135\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 41.4558i 1.82676i
\(516\) 0 0
\(517\) − 28.2960i − 1.24446i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −3.93562 −0.172423 −0.0862113 0.996277i \(-0.527476\pi\)
−0.0862113 + 0.996277i \(0.527476\pi\)
\(522\) 0 0
\(523\) − 31.7289i − 1.38741i −0.720260 0.693705i \(-0.755977\pi\)
0.720260 0.693705i \(-0.244023\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 41.4558i − 1.80584i
\(528\) 0 0
\(529\) 19.0000 0.826087
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 3.55635i − 0.154043i
\(534\) 0 0
\(535\) − 39.0112i − 1.68660i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −14.9706 −0.643635 −0.321817 0.946802i \(-0.604294\pi\)
−0.321817 + 0.946802i \(0.604294\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −15.8645 −0.679559
\(546\) 0 0
\(547\) −18.6274 −0.796451 −0.398225 0.917288i \(-0.630374\pi\)
−0.398225 + 0.917288i \(0.630374\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 4.85483 0.206823
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 13.4142i − 0.568378i −0.958768 0.284189i \(-0.908276\pi\)
0.958768 0.284189i \(-0.0917244\pi\)
\(558\) 0 0
\(559\) − 13.1426i − 0.555871i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 14.1480 0.596268 0.298134 0.954524i \(-0.403636\pi\)
0.298134 + 0.954524i \(0.403636\pi\)
\(564\) 0 0
\(565\) 40.7276i 1.71342i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 37.1127i − 1.55585i −0.628360 0.777923i \(-0.716274\pi\)
0.628360 0.777923i \(-0.283726\pi\)
\(570\) 0 0
\(571\) −11.3137 −0.473464 −0.236732 0.971575i \(-0.576076\pi\)
−0.236732 + 0.971575i \(0.576076\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 7.17157i − 0.299075i
\(576\) 0 0
\(577\) − 29.5098i − 1.22851i −0.789109 0.614254i \(-0.789457\pi\)
0.789109 0.614254i \(-0.210543\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −34.1421 −1.41402
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 17.5809 0.725642 0.362821 0.931859i \(-0.381814\pi\)
0.362821 + 0.931859i \(0.381814\pi\)
\(588\) 0 0
\(589\) 34.3431 1.41508
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 23.6494 0.971166 0.485583 0.874190i \(-0.338607\pi\)
0.485583 + 0.874190i \(0.338607\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 10.4853i 0.428417i 0.976788 + 0.214208i \(0.0687172\pi\)
−0.976788 + 0.214208i \(0.931283\pi\)
\(600\) 0 0
\(601\) 40.5194i 1.65282i 0.563069 + 0.826410i \(0.309621\pi\)
−0.563069 + 0.826410i \(0.690379\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −36.0810 −1.46690
\(606\) 0 0
\(607\) − 3.43289i − 0.139337i −0.997570 0.0696683i \(-0.977806\pi\)
0.997570 0.0696683i \(-0.0221941\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 17.1716i − 0.694687i
\(612\) 0 0
\(613\) 42.8701 1.73151 0.865753 0.500472i \(-0.166840\pi\)
0.865753 + 0.500472i \(0.166840\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 13.5147i − 0.544082i −0.962286 0.272041i \(-0.912301\pi\)
0.962286 0.272041i \(-0.0876986\pi\)
\(618\) 0 0
\(619\) 44.8715i 1.80354i 0.432219 + 0.901769i \(0.357731\pi\)
−0.432219 + 0.901769i \(0.642269\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −30.0711 −1.20284
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 38.3002 1.52713
\(630\) 0 0
\(631\) 26.8284 1.06802 0.534011 0.845477i \(-0.320684\pi\)
0.534011 + 0.845477i \(0.320684\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −48.3044 −1.91690
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 24.4853i 0.967110i 0.875314 + 0.483555i \(0.160655\pi\)
−0.875314 + 0.483555i \(0.839345\pi\)
\(642\) 0 0
\(643\) − 5.86030i − 0.231108i −0.993301 0.115554i \(-0.963136\pi\)
0.993301 0.115554i \(-0.0368643\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 10.7151 0.421255 0.210628 0.977566i \(-0.432449\pi\)
0.210628 + 0.977566i \(0.432449\pi\)
\(648\) 0 0
\(649\) − 28.2960i − 1.11072i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 21.5147i − 0.841936i −0.907075 0.420968i \(-0.861690\pi\)
0.907075 0.420968i \(-0.138310\pi\)
\(654\) 0 0
\(655\) −24.2843 −0.948865
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 3.45584i − 0.134621i −0.997732 0.0673103i \(-0.978558\pi\)
0.997732 0.0673103i \(-0.0214417\pi\)
\(660\) 0 0
\(661\) 22.9385i 0.892203i 0.894982 + 0.446102i \(0.147188\pi\)
−0.894982 + 0.446102i \(0.852812\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1.65685 0.0641537
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −5.86030 −0.226234
\(672\) 0 0
\(673\) −18.1005 −0.697723 −0.348862 0.937174i \(-0.613432\pi\)
−0.348862 + 0.937174i \(0.613432\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 46.0852 1.77120 0.885599 0.464451i \(-0.153748\pi\)
0.885599 + 0.464451i \(0.153748\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 34.7696i 1.33042i 0.746656 + 0.665210i \(0.231658\pi\)
−0.746656 + 0.665210i \(0.768342\pi\)
\(684\) 0 0
\(685\) 14.1480i 0.540568i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −20.7193 −0.789342
\(690\) 0 0
\(691\) − 8.28772i − 0.315280i −0.987497 0.157640i \(-0.949611\pi\)
0.987497 0.157640i \(-0.0503885\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 24.2843i − 0.921155i
\(696\) 0 0
\(697\) 8.58579 0.325210
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 32.7696i 1.23769i 0.785514 + 0.618844i \(0.212399\pi\)
−0.785514 + 0.618844i \(0.787601\pi\)
\(702\) 0 0
\(703\) 31.7289i 1.19668i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −7.27208 −0.273109 −0.136554 0.990633i \(-0.543603\pi\)
−0.136554 + 0.990633i \(0.543603\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 11.7206 0.438940
\(714\) 0 0
\(715\) −41.4558 −1.55036
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 11.7206 0.437105 0.218552 0.975825i \(-0.429867\pi\)
0.218552 + 0.975825i \(0.429867\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 2.97056i 0.110324i
\(726\) 0 0
\(727\) 21.0138i 0.779358i 0.920951 + 0.389679i \(0.127414\pi\)
−0.920951 + 0.389679i \(0.872586\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 31.7289 1.17354
\(732\) 0 0
\(733\) 37.7975i 1.39608i 0.716058 + 0.698041i \(0.245945\pi\)
−0.716058 + 0.698041i \(0.754055\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 40.9706i 1.50917i
\(738\) 0 0
\(739\) 13.1716 0.484524 0.242262 0.970211i \(-0.422111\pi\)
0.242262 + 0.970211i \(0.422111\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 32.4264i 1.18961i 0.803870 + 0.594805i \(0.202771\pi\)
−0.803870 + 0.594805i \(0.797229\pi\)
\(744\) 0 0
\(745\) − 12.4316i − 0.455458i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −43.3137 −1.58054 −0.790270 0.612759i \(-0.790060\pi\)
−0.790270 + 0.612759i \(0.790060\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −13.1426 −0.478306
\(756\) 0 0
\(757\) −46.8701 −1.70352 −0.851761 0.523931i \(-0.824465\pi\)
−0.851761 + 0.523931i \(0.824465\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −17.0782 −0.619083 −0.309542 0.950886i \(-0.600176\pi\)
−0.309542 + 0.950886i \(0.600176\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 17.1716i − 0.620030i
\(768\) 0 0
\(769\) − 48.0961i − 1.73439i −0.497968 0.867195i \(-0.665920\pi\)
0.497968 0.867195i \(-0.334080\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 24.6549 0.886776 0.443388 0.896330i \(-0.353776\pi\)
0.443388 + 0.896330i \(0.353776\pi\)
\(774\) 0 0
\(775\) 21.0138i 0.754838i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 7.11270i 0.254839i
\(780\) 0 0
\(781\) −4.00000 −0.143131
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 67.2132i − 2.39894i
\(786\) 0 0
\(787\) 11.7206i 0.417794i 0.977938 + 0.208897i \(0.0669874\pi\)
−0.977938 + 0.208897i \(0.933013\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −3.55635 −0.126290
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −51.2345 −1.81482 −0.907410 0.420247i \(-0.861944\pi\)
−0.907410 + 0.420247i \(0.861944\pi\)
\(798\) 0 0
\(799\) 41.4558 1.46660
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −34.1563 −1.20535
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 16.0416i − 0.563994i −0.959415 0.281997i \(-0.909003\pi\)
0.959415 0.281997i \(-0.0909968\pi\)
\(810\) 0 0
\(811\) 8.28772i 0.291021i 0.989357 + 0.145511i \(0.0464825\pi\)
−0.989357 + 0.145511i \(0.953518\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −26.8741 −0.941359
\(816\) 0 0
\(817\) 26.2851i 0.919599i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 23.7574i 0.829138i 0.910018 + 0.414569i \(0.136068\pi\)
−0.910018 + 0.414569i \(0.863932\pi\)
\(822\) 0 0
\(823\) 41.9411 1.46198 0.730988 0.682390i \(-0.239060\pi\)
0.730988 + 0.682390i \(0.239060\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 39.6569i − 1.37900i −0.724284 0.689502i \(-0.757829\pi\)
0.724284 0.689502i \(-0.242171\pi\)
\(828\) 0 0
\(829\) 36.0810i 1.25315i 0.779363 + 0.626573i \(0.215543\pi\)
−0.779363 + 0.626573i \(0.784457\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 41.4558 1.43464
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 17.5809 0.606960 0.303480 0.952838i \(-0.401851\pi\)
0.303480 + 0.952838i \(0.401851\pi\)
\(840\) 0 0
\(841\) 28.3137 0.976335
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 12.9343 0.444954
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 10.8284i 0.371194i
\(852\) 0 0
\(853\) − 16.0727i − 0.550319i −0.961399 0.275159i \(-0.911269\pi\)
0.961399 0.275159i \(-0.0887306\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −0.502734 −0.0171731 −0.00858654 0.999963i \(-0.502733\pi\)
−0.00858654 + 0.999963i \(0.502733\pi\)
\(858\) 0 0
\(859\) 9.29319i 0.317079i 0.987353 + 0.158540i \(0.0506786\pi\)
−0.987353 + 0.158540i \(0.949321\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 10.6863i 0.363766i 0.983320 + 0.181883i \(0.0582191\pi\)
−0.983320 + 0.181883i \(0.941781\pi\)
\(864\) 0 0
\(865\) −3.55635 −0.120919
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 8.00000i − 0.271381i
\(870\) 0 0
\(871\) 24.8632i 0.842456i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −32.5269 −1.09836 −0.549178 0.835705i \(-0.685059\pi\)
−0.549178 + 0.835705i \(0.685059\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −23.6494 −0.796770 −0.398385 0.917218i \(-0.630429\pi\)
−0.398385 + 0.917218i \(0.630429\pi\)
\(882\) 0 0
\(883\) −13.4558 −0.452825 −0.226413 0.974031i \(-0.572700\pi\)
−0.226413 + 0.974031i \(0.572700\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 50.7318 1.70341 0.851703 0.524024i \(-0.175570\pi\)
0.851703 + 0.524024i \(0.175570\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 34.3431i 1.14925i
\(894\) 0 0
\(895\) 77.6059i 2.59408i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −4.85483 −0.161918
\(900\) 0 0
\(901\) − 50.0208i − 1.66643i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 27.8406i 0.925454i
\(906\) 0 0
\(907\) −28.7696 −0.955277 −0.477639 0.878556i \(-0.658507\pi\)
−0.477639 + 0.878556i \(0.658507\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 42.4853i − 1.40760i −0.710398 0.703800i \(-0.751485\pi\)
0.710398 0.703800i \(-0.248515\pi\)
\(912\) 0 0
\(913\) 56.5921i 1.87292i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 41.4558 1.36750 0.683751 0.729715i \(-0.260347\pi\)
0.683751 + 0.729715i \(0.260347\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −2.42742 −0.0798994
\(924\) 0 0
\(925\) −19.4142 −0.638335
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −0.502734 −0.0164942 −0.00824709 0.999966i \(-0.502625\pi\)
−0.00824709 + 0.999966i \(0.502625\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) − 100.083i − 3.27307i
\(936\) 0 0
\(937\) 51.2345i 1.67376i 0.547387 + 0.836879i \(0.315622\pi\)
−0.547387 + 0.836879i \(0.684378\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 4.64659 0.151475 0.0757373 0.997128i \(-0.475869\pi\)
0.0757373 + 0.997128i \(0.475869\pi\)
\(942\) 0 0
\(943\) 2.42742i 0.0790476i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 10.9706i 0.356495i 0.983986 + 0.178248i \(0.0570428\pi\)
−0.983986 + 0.178248i \(0.942957\pi\)
\(948\) 0 0
\(949\) −20.7279 −0.672857
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 10.3848i 0.336396i 0.985753 + 0.168198i \(0.0537948\pi\)
−0.985753 + 0.168198i \(0.946205\pi\)
\(954\) 0 0
\(955\) 44.4550i 1.43853i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −3.34315 −0.107843
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −18.5864 −0.598317
\(966\) 0 0
\(967\) 19.1127 0.614623 0.307311 0.951609i \(-0.400571\pi\)
0.307311 + 0.951609i \(0.400571\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 4.85483 0.155799 0.0778995 0.996961i \(-0.475179\pi\)
0.0778995 + 0.996961i \(0.475179\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 44.8284i 1.43419i 0.696976 + 0.717094i \(0.254528\pi\)
−0.696976 + 0.717094i \(0.745472\pi\)
\(978\) 0 0
\(979\) − 54.1647i − 1.73111i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −31.7289 −1.01200 −0.505998 0.862535i \(-0.668876\pi\)
−0.505998 + 0.862535i \(0.668876\pi\)
\(984\) 0 0
\(985\) 22.7302i 0.724246i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 8.97056i 0.285247i
\(990\) 0 0
\(991\) 59.3970 1.88681 0.943403 0.331647i \(-0.107604\pi\)
0.943403 + 0.331647i \(0.107604\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 48.5685i − 1.53973i
\(996\) 0 0
\(997\) − 39.5139i − 1.25142i −0.780056 0.625709i \(-0.784810\pi\)
0.780056 0.625709i \(-0.215190\pi\)
\(998\) 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.881.6 8
3.2 odd 2 inner 7056.2.k.g.881.3 8
4.3 odd 2 441.2.c.b.440.4 yes 8
7.6 odd 2 inner 7056.2.k.g.881.4 8
12.11 even 2 441.2.c.b.440.5 yes 8
21.20 even 2 inner 7056.2.k.g.881.5 8
28.3 even 6 441.2.p.c.215.6 16
28.11 odd 6 441.2.p.c.215.5 16
28.19 even 6 441.2.p.c.80.4 16
28.23 odd 6 441.2.p.c.80.3 16
28.27 even 2 441.2.c.b.440.3 8
84.11 even 6 441.2.p.c.215.4 16
84.23 even 6 441.2.p.c.80.6 16
84.47 odd 6 441.2.p.c.80.5 16
84.59 odd 6 441.2.p.c.215.3 16
84.83 odd 2 441.2.c.b.440.6 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
441.2.c.b.440.3 8 28.27 even 2
441.2.c.b.440.4 yes 8 4.3 odd 2
441.2.c.b.440.5 yes 8 12.11 even 2
441.2.c.b.440.6 yes 8 84.83 odd 2
441.2.p.c.80.3 16 28.23 odd 6
441.2.p.c.80.4 16 28.19 even 6
441.2.p.c.80.5 16 84.47 odd 6
441.2.p.c.80.6 16 84.23 even 6
441.2.p.c.215.3 16 84.59 odd 6
441.2.p.c.215.4 16 84.11 even 6
441.2.p.c.215.5 16 28.11 odd 6
441.2.p.c.215.6 16 28.3 even 6
7056.2.k.g.881.3 8 3.2 odd 2 inner
7056.2.k.g.881.4 8 7.6 odd 2 inner
7056.2.k.g.881.5 8 21.20 even 2 inner
7056.2.k.g.881.6 8 1.1 even 1 trivial