Properties

Label 9360.2.a.ch.1.2
Level $9360$
Weight $2$
Character 9360.1
Self dual yes
Analytic conductor $74.740$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9360 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9360.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(74.7399762919\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 390)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 9360.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{5} +2.82843 q^{7} +O(q^{10})\) \(q-1.00000 q^{5} +2.82843 q^{7} +5.65685 q^{11} -1.00000 q^{13} -0.828427 q^{17} -2.82843 q^{19} -8.48528 q^{23} +1.00000 q^{25} +8.82843 q^{29} -4.00000 q^{31} -2.82843 q^{35} -11.6569 q^{37} +7.65685 q^{41} -9.65685 q^{43} -8.00000 q^{47} +1.00000 q^{49} -13.3137 q^{53} -5.65685 q^{55} -2.34315 q^{59} +6.00000 q^{61} +1.00000 q^{65} -5.65685 q^{67} -5.65685 q^{71} -14.4853 q^{73} +16.0000 q^{77} -2.34315 q^{79} +6.34315 q^{83} +0.828427 q^{85} +15.6569 q^{89} -2.82843 q^{91} +2.82843 q^{95} -3.17157 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{5} + O(q^{10}) \) \( 2q - 2q^{5} - 2q^{13} + 4q^{17} + 2q^{25} + 12q^{29} - 8q^{31} - 12q^{37} + 4q^{41} - 8q^{43} - 16q^{47} + 2q^{49} - 4q^{53} - 16q^{59} + 12q^{61} + 2q^{65} - 12q^{73} + 32q^{77} - 16q^{79} + 24q^{83} - 4q^{85} + 20q^{89} - 12q^{97} + O(q^{100}) \)

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
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 2.82843 1.06904 0.534522 0.845154i \(-0.320491\pi\)
0.534522 + 0.845154i \(0.320491\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.65685 1.70561 0.852803 0.522233i \(-0.174901\pi\)
0.852803 + 0.522233i \(0.174901\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.828427 −0.200923 −0.100462 0.994941i \(-0.532032\pi\)
−0.100462 + 0.994941i \(0.532032\pi\)
\(18\) 0 0
\(19\) −2.82843 −0.648886 −0.324443 0.945905i \(-0.605177\pi\)
−0.324443 + 0.945905i \(0.605177\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −8.48528 −1.76930 −0.884652 0.466252i \(-0.845604\pi\)
−0.884652 + 0.466252i \(0.845604\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 8.82843 1.63940 0.819699 0.572795i \(-0.194141\pi\)
0.819699 + 0.572795i \(0.194141\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.82843 −0.478091
\(36\) 0 0
\(37\) −11.6569 −1.91638 −0.958188 0.286141i \(-0.907627\pi\)
−0.958188 + 0.286141i \(0.907627\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 7.65685 1.19580 0.597900 0.801571i \(-0.296002\pi\)
0.597900 + 0.801571i \(0.296002\pi\)
\(42\) 0 0
\(43\) −9.65685 −1.47266 −0.736328 0.676625i \(-0.763442\pi\)
−0.736328 + 0.676625i \(0.763442\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −13.3137 −1.82878 −0.914389 0.404836i \(-0.867329\pi\)
−0.914389 + 0.404836i \(0.867329\pi\)
\(54\) 0 0
\(55\) −5.65685 −0.762770
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2.34315 −0.305052 −0.152526 0.988299i \(-0.548741\pi\)
−0.152526 + 0.988299i \(0.548741\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.00000 0.124035
\(66\) 0 0
\(67\) −5.65685 −0.691095 −0.345547 0.938401i \(-0.612307\pi\)
−0.345547 + 0.938401i \(0.612307\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −5.65685 −0.671345 −0.335673 0.941979i \(-0.608964\pi\)
−0.335673 + 0.941979i \(0.608964\pi\)
\(72\) 0 0
\(73\) −14.4853 −1.69537 −0.847687 0.530497i \(-0.822005\pi\)
−0.847687 + 0.530497i \(0.822005\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 16.0000 1.82337
\(78\) 0 0
\(79\) −2.34315 −0.263624 −0.131812 0.991275i \(-0.542080\pi\)
−0.131812 + 0.991275i \(0.542080\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 6.34315 0.696251 0.348125 0.937448i \(-0.386818\pi\)
0.348125 + 0.937448i \(0.386818\pi\)
\(84\) 0 0
\(85\) 0.828427 0.0898555
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 15.6569 1.65962 0.829812 0.558044i \(-0.188448\pi\)
0.829812 + 0.558044i \(0.188448\pi\)
\(90\) 0 0
\(91\) −2.82843 −0.296500
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.82843 0.290191
\(96\) 0 0
\(97\) −3.17157 −0.322024 −0.161012 0.986952i \(-0.551476\pi\)
−0.161012 + 0.986952i \(0.551476\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −16.1421 −1.60620 −0.803101 0.595843i \(-0.796818\pi\)
−0.803101 + 0.595843i \(0.796818\pi\)
\(102\) 0 0
\(103\) 1.65685 0.163255 0.0816274 0.996663i \(-0.473988\pi\)
0.0816274 + 0.996663i \(0.473988\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 0 0
\(109\) 8.82843 0.845610 0.422805 0.906221i \(-0.361046\pi\)
0.422805 + 0.906221i \(0.361046\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −6.48528 −0.610084 −0.305042 0.952339i \(-0.598670\pi\)
−0.305042 + 0.952339i \(0.598670\pi\)
\(114\) 0 0
\(115\) 8.48528 0.791257
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −2.34315 −0.214796
\(120\) 0 0
\(121\) 21.0000 1.90909
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −9.65685 −0.856907 −0.428454 0.903564i \(-0.640941\pi\)
−0.428454 + 0.903564i \(0.640941\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −6.14214 −0.536641 −0.268320 0.963330i \(-0.586469\pi\)
−0.268320 + 0.963330i \(0.586469\pi\)
\(132\) 0 0
\(133\) −8.00000 −0.693688
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 17.3137 1.47921 0.739605 0.673041i \(-0.235012\pi\)
0.739605 + 0.673041i \(0.235012\pi\)
\(138\) 0 0
\(139\) −6.34315 −0.538019 −0.269009 0.963138i \(-0.586696\pi\)
−0.269009 + 0.963138i \(0.586696\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −5.65685 −0.473050
\(144\) 0 0
\(145\) −8.82843 −0.733161
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −3.65685 −0.299581 −0.149791 0.988718i \(-0.547860\pi\)
−0.149791 + 0.988718i \(0.547860\pi\)
\(150\) 0 0
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 4.00000 0.321288
\(156\) 0 0
\(157\) 5.31371 0.424080 0.212040 0.977261i \(-0.431989\pi\)
0.212040 + 0.977261i \(0.431989\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −24.0000 −1.89146
\(162\) 0 0
\(163\) −11.3137 −0.886158 −0.443079 0.896483i \(-0.646114\pi\)
−0.443079 + 0.896483i \(0.646114\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 8.97056 0.694163 0.347081 0.937835i \(-0.387173\pi\)
0.347081 + 0.937835i \(0.387173\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 9.31371 0.708108 0.354054 0.935225i \(-0.384803\pi\)
0.354054 + 0.935225i \(0.384803\pi\)
\(174\) 0 0
\(175\) 2.82843 0.213809
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 7.51472 0.561676 0.280838 0.959755i \(-0.409388\pi\)
0.280838 + 0.959755i \(0.409388\pi\)
\(180\) 0 0
\(181\) −7.65685 −0.569129 −0.284565 0.958657i \(-0.591849\pi\)
−0.284565 + 0.958657i \(0.591849\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 11.6569 0.857029
\(186\) 0 0
\(187\) −4.68629 −0.342696
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 11.3137 0.818631 0.409316 0.912393i \(-0.365768\pi\)
0.409316 + 0.912393i \(0.365768\pi\)
\(192\) 0 0
\(193\) 2.48528 0.178894 0.0894472 0.995992i \(-0.471490\pi\)
0.0894472 + 0.995992i \(0.471490\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 13.3137 0.948562 0.474281 0.880373i \(-0.342708\pi\)
0.474281 + 0.880373i \(0.342708\pi\)
\(198\) 0 0
\(199\) −10.3431 −0.733206 −0.366603 0.930377i \(-0.619479\pi\)
−0.366603 + 0.930377i \(0.619479\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 24.9706 1.75259
\(204\) 0 0
\(205\) −7.65685 −0.534778
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) −0.686292 −0.0472463 −0.0236231 0.999721i \(-0.507520\pi\)
−0.0236231 + 0.999721i \(0.507520\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 9.65685 0.658592
\(216\) 0 0
\(217\) −11.3137 −0.768025
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0.828427 0.0557260
\(222\) 0 0
\(223\) −10.8284 −0.725125 −0.362563 0.931959i \(-0.618098\pi\)
−0.362563 + 0.931959i \(0.618098\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 4.00000 0.265489 0.132745 0.991150i \(-0.457621\pi\)
0.132745 + 0.991150i \(0.457621\pi\)
\(228\) 0 0
\(229\) 4.14214 0.273720 0.136860 0.990590i \(-0.456299\pi\)
0.136860 + 0.990590i \(0.456299\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −5.51472 −0.361281 −0.180641 0.983549i \(-0.557817\pi\)
−0.180641 + 0.983549i \(0.557817\pi\)
\(234\) 0 0
\(235\) 8.00000 0.521862
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 16.0000 1.03495 0.517477 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 0 0
\(241\) 5.31371 0.342286 0.171143 0.985246i \(-0.445254\pi\)
0.171143 + 0.985246i \(0.445254\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 2.82843 0.179969
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 10.8284 0.683484 0.341742 0.939794i \(-0.388983\pi\)
0.341742 + 0.939794i \(0.388983\pi\)
\(252\) 0 0
\(253\) −48.0000 −3.01773
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 4.82843 0.301189 0.150595 0.988596i \(-0.451881\pi\)
0.150595 + 0.988596i \(0.451881\pi\)
\(258\) 0 0
\(259\) −32.9706 −2.04869
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 16.4853 1.01653 0.508263 0.861202i \(-0.330288\pi\)
0.508263 + 0.861202i \(0.330288\pi\)
\(264\) 0 0
\(265\) 13.3137 0.817855
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 14.4853 0.883183 0.441592 0.897216i \(-0.354414\pi\)
0.441592 + 0.897216i \(0.354414\pi\)
\(270\) 0 0
\(271\) −7.31371 −0.444276 −0.222138 0.975015i \(-0.571304\pi\)
−0.222138 + 0.975015i \(0.571304\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 5.65685 0.341121
\(276\) 0 0
\(277\) 26.0000 1.56219 0.781094 0.624413i \(-0.214662\pi\)
0.781094 + 0.624413i \(0.214662\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −8.34315 −0.497710 −0.248855 0.968541i \(-0.580054\pi\)
−0.248855 + 0.968541i \(0.580054\pi\)
\(282\) 0 0
\(283\) 17.6569 1.04959 0.524796 0.851228i \(-0.324142\pi\)
0.524796 + 0.851228i \(0.324142\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 21.6569 1.27836
\(288\) 0 0
\(289\) −16.3137 −0.959630
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 16.6274 0.971384 0.485692 0.874130i \(-0.338568\pi\)
0.485692 + 0.874130i \(0.338568\pi\)
\(294\) 0 0
\(295\) 2.34315 0.136423
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 8.48528 0.490716
\(300\) 0 0
\(301\) −27.3137 −1.57434
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −6.00000 −0.343559
\(306\) 0 0
\(307\) 21.6569 1.23602 0.618011 0.786169i \(-0.287939\pi\)
0.618011 + 0.786169i \(0.287939\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) −30.9706 −1.75056 −0.875280 0.483617i \(-0.839323\pi\)
−0.875280 + 0.483617i \(0.839323\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −25.3137 −1.42176 −0.710880 0.703314i \(-0.751703\pi\)
−0.710880 + 0.703314i \(0.751703\pi\)
\(318\) 0 0
\(319\) 49.9411 2.79617
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 2.34315 0.130376
\(324\) 0 0
\(325\) −1.00000 −0.0554700
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −22.6274 −1.24749
\(330\) 0 0
\(331\) 8.48528 0.466393 0.233197 0.972430i \(-0.425081\pi\)
0.233197 + 0.972430i \(0.425081\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 5.65685 0.309067
\(336\) 0 0
\(337\) 10.9706 0.597605 0.298802 0.954315i \(-0.403413\pi\)
0.298802 + 0.954315i \(0.403413\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −22.6274 −1.22534
\(342\) 0 0
\(343\) −16.9706 −0.916324
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −9.65685 −0.518407 −0.259204 0.965823i \(-0.583460\pi\)
−0.259204 + 0.965823i \(0.583460\pi\)
\(348\) 0 0
\(349\) 12.1421 0.649954 0.324977 0.945722i \(-0.394644\pi\)
0.324977 + 0.945722i \(0.394644\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −5.31371 −0.282820 −0.141410 0.989951i \(-0.545164\pi\)
−0.141410 + 0.989951i \(0.545164\pi\)
\(354\) 0 0
\(355\) 5.65685 0.300235
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −28.2843 −1.49279 −0.746393 0.665505i \(-0.768216\pi\)
−0.746393 + 0.665505i \(0.768216\pi\)
\(360\) 0 0
\(361\) −11.0000 −0.578947
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 14.4853 0.758194
\(366\) 0 0
\(367\) −25.6569 −1.33928 −0.669638 0.742687i \(-0.733551\pi\)
−0.669638 + 0.742687i \(0.733551\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −37.6569 −1.95505
\(372\) 0 0
\(373\) −2.68629 −0.139091 −0.0695455 0.997579i \(-0.522155\pi\)
−0.0695455 + 0.997579i \(0.522155\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −8.82843 −0.454687
\(378\) 0 0
\(379\) 7.51472 0.386005 0.193003 0.981198i \(-0.438177\pi\)
0.193003 + 0.981198i \(0.438177\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −29.6569 −1.51539 −0.757697 0.652606i \(-0.773676\pi\)
−0.757697 + 0.652606i \(0.773676\pi\)
\(384\) 0 0
\(385\) −16.0000 −0.815436
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 6.48528 0.328817 0.164408 0.986392i \(-0.447429\pi\)
0.164408 + 0.986392i \(0.447429\pi\)
\(390\) 0 0
\(391\) 7.02944 0.355494
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 2.34315 0.117896
\(396\) 0 0
\(397\) 30.2843 1.51992 0.759962 0.649968i \(-0.225218\pi\)
0.759962 + 0.649968i \(0.225218\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 26.9706 1.34685 0.673423 0.739258i \(-0.264823\pi\)
0.673423 + 0.739258i \(0.264823\pi\)
\(402\) 0 0
\(403\) 4.00000 0.199254
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −65.9411 −3.26858
\(408\) 0 0
\(409\) −3.65685 −0.180820 −0.0904099 0.995905i \(-0.528818\pi\)
−0.0904099 + 0.995905i \(0.528818\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −6.62742 −0.326114
\(414\) 0 0
\(415\) −6.34315 −0.311373
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −10.8284 −0.529003 −0.264502 0.964385i \(-0.585207\pi\)
−0.264502 + 0.964385i \(0.585207\pi\)
\(420\) 0 0
\(421\) −24.1421 −1.17662 −0.588308 0.808637i \(-0.700206\pi\)
−0.588308 + 0.808637i \(0.700206\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −0.828427 −0.0401846
\(426\) 0 0
\(427\) 16.9706 0.821263
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −16.0000 −0.770693 −0.385346 0.922772i \(-0.625918\pi\)
−0.385346 + 0.922772i \(0.625918\pi\)
\(432\) 0 0
\(433\) −22.9706 −1.10389 −0.551947 0.833879i \(-0.686115\pi\)
−0.551947 + 0.833879i \(0.686115\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 24.0000 1.14808
\(438\) 0 0
\(439\) −22.6274 −1.07995 −0.539974 0.841682i \(-0.681566\pi\)
−0.539974 + 0.841682i \(0.681566\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 30.3431 1.44165 0.720823 0.693119i \(-0.243764\pi\)
0.720823 + 0.693119i \(0.243764\pi\)
\(444\) 0 0
\(445\) −15.6569 −0.742206
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −26.2843 −1.24043 −0.620216 0.784431i \(-0.712955\pi\)
−0.620216 + 0.784431i \(0.712955\pi\)
\(450\) 0 0
\(451\) 43.3137 2.03956
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 2.82843 0.132599
\(456\) 0 0
\(457\) 20.8284 0.974313 0.487156 0.873315i \(-0.338034\pi\)
0.487156 + 0.873315i \(0.338034\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −14.0000 −0.652045 −0.326023 0.945362i \(-0.605709\pi\)
−0.326023 + 0.945362i \(0.605709\pi\)
\(462\) 0 0
\(463\) 3.79899 0.176554 0.0882770 0.996096i \(-0.471864\pi\)
0.0882770 + 0.996096i \(0.471864\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 7.31371 0.338438 0.169219 0.985578i \(-0.445875\pi\)
0.169219 + 0.985578i \(0.445875\pi\)
\(468\) 0 0
\(469\) −16.0000 −0.738811
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −54.6274 −2.51177
\(474\) 0 0
\(475\) −2.82843 −0.129777
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −11.3137 −0.516937 −0.258468 0.966020i \(-0.583218\pi\)
−0.258468 + 0.966020i \(0.583218\pi\)
\(480\) 0 0
\(481\) 11.6569 0.531507
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 3.17157 0.144014
\(486\) 0 0
\(487\) −16.4853 −0.747019 −0.373510 0.927626i \(-0.621846\pi\)
−0.373510 + 0.927626i \(0.621846\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −38.1421 −1.72133 −0.860665 0.509171i \(-0.829952\pi\)
−0.860665 + 0.509171i \(0.829952\pi\)
\(492\) 0 0
\(493\) −7.31371 −0.329393
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −16.0000 −0.717698
\(498\) 0 0
\(499\) −0.485281 −0.0217242 −0.0108621 0.999941i \(-0.503458\pi\)
−0.0108621 + 0.999941i \(0.503458\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −23.5147 −1.04847 −0.524235 0.851574i \(-0.675649\pi\)
−0.524235 + 0.851574i \(0.675649\pi\)
\(504\) 0 0
\(505\) 16.1421 0.718316
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 37.3137 1.65390 0.826951 0.562275i \(-0.190074\pi\)
0.826951 + 0.562275i \(0.190074\pi\)
\(510\) 0 0
\(511\) −40.9706 −1.81243
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −1.65685 −0.0730097
\(516\) 0 0
\(517\) −45.2548 −1.99031
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −26.9706 −1.18160 −0.590801 0.806817i \(-0.701188\pi\)
−0.590801 + 0.806817i \(0.701188\pi\)
\(522\) 0 0
\(523\) 10.6274 0.464704 0.232352 0.972632i \(-0.425358\pi\)
0.232352 + 0.972632i \(0.425358\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 3.31371 0.144347
\(528\) 0 0
\(529\) 49.0000 2.13043
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −7.65685 −0.331655
\(534\) 0 0
\(535\) 4.00000 0.172935
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 5.65685 0.243658
\(540\) 0 0
\(541\) 14.4853 0.622771 0.311385 0.950284i \(-0.399207\pi\)
0.311385 + 0.950284i \(0.399207\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −8.82843 −0.378168
\(546\) 0 0
\(547\) −0.686292 −0.0293437 −0.0146719 0.999892i \(-0.504670\pi\)
−0.0146719 + 0.999892i \(0.504670\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −24.9706 −1.06378
\(552\) 0 0
\(553\) −6.62742 −0.281826
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −10.6863 −0.452793 −0.226396 0.974035i \(-0.572694\pi\)
−0.226396 + 0.974035i \(0.572694\pi\)
\(558\) 0 0
\(559\) 9.65685 0.408441
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −30.3431 −1.27881 −0.639406 0.768870i \(-0.720819\pi\)
−0.639406 + 0.768870i \(0.720819\pi\)
\(564\) 0 0
\(565\) 6.48528 0.272838
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −31.6569 −1.32712 −0.663562 0.748121i \(-0.730956\pi\)
−0.663562 + 0.748121i \(0.730956\pi\)
\(570\) 0 0
\(571\) −20.9706 −0.877591 −0.438795 0.898587i \(-0.644595\pi\)
−0.438795 + 0.898587i \(0.644595\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −8.48528 −0.353861
\(576\) 0 0
\(577\) −23.4558 −0.976480 −0.488240 0.872710i \(-0.662361\pi\)
−0.488240 + 0.872710i \(0.662361\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 17.9411 0.744323
\(582\) 0 0
\(583\) −75.3137 −3.11918
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 2.62742 0.108445 0.0542226 0.998529i \(-0.482732\pi\)
0.0542226 + 0.998529i \(0.482732\pi\)
\(588\) 0 0
\(589\) 11.3137 0.466173
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0.343146 0.0140913 0.00704565 0.999975i \(-0.497757\pi\)
0.00704565 + 0.999975i \(0.497757\pi\)
\(594\) 0 0
\(595\) 2.34315 0.0960596
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 40.0000 1.63436 0.817178 0.576386i \(-0.195537\pi\)
0.817178 + 0.576386i \(0.195537\pi\)
\(600\) 0 0
\(601\) 29.3137 1.19573 0.597866 0.801596i \(-0.296016\pi\)
0.597866 + 0.801596i \(0.296016\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −21.0000 −0.853771
\(606\) 0 0
\(607\) −28.9706 −1.17588 −0.587939 0.808905i \(-0.700061\pi\)
−0.587939 + 0.808905i \(0.700061\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 8.00000 0.323645
\(612\) 0 0
\(613\) 22.2843 0.900053 0.450027 0.893015i \(-0.351414\pi\)
0.450027 + 0.893015i \(0.351414\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −2.00000 −0.0805170 −0.0402585 0.999189i \(-0.512818\pi\)
−0.0402585 + 0.999189i \(0.512818\pi\)
\(618\) 0 0
\(619\) 34.8284 1.39987 0.699936 0.714205i \(-0.253212\pi\)
0.699936 + 0.714205i \(0.253212\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 44.2843 1.77421
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 9.65685 0.385044
\(630\) 0 0
\(631\) 33.6569 1.33986 0.669929 0.742425i \(-0.266324\pi\)
0.669929 + 0.742425i \(0.266324\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 9.65685 0.383221
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 4.62742 0.182772 0.0913860 0.995816i \(-0.470870\pi\)
0.0913860 + 0.995816i \(0.470870\pi\)
\(642\) 0 0
\(643\) −39.5980 −1.56159 −0.780796 0.624786i \(-0.785186\pi\)
−0.780796 + 0.624786i \(0.785186\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 8.48528 0.333591 0.166795 0.985992i \(-0.446658\pi\)
0.166795 + 0.985992i \(0.446658\pi\)
\(648\) 0 0
\(649\) −13.2548 −0.520298
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 42.2843 1.65471 0.827356 0.561678i \(-0.189844\pi\)
0.827356 + 0.561678i \(0.189844\pi\)
\(654\) 0 0
\(655\) 6.14214 0.239993
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −7.51472 −0.292732 −0.146366 0.989231i \(-0.546758\pi\)
−0.146366 + 0.989231i \(0.546758\pi\)
\(660\) 0 0
\(661\) −8.14214 −0.316692 −0.158346 0.987384i \(-0.550616\pi\)
−0.158346 + 0.987384i \(0.550616\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 8.00000 0.310227
\(666\) 0 0
\(667\) −74.9117 −2.90059
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 33.9411 1.31028
\(672\) 0 0
\(673\) 32.6274 1.25769 0.628847 0.777529i \(-0.283527\pi\)
0.628847 + 0.777529i \(0.283527\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −12.3431 −0.474386 −0.237193 0.971463i \(-0.576227\pi\)
−0.237193 + 0.971463i \(0.576227\pi\)
\(678\) 0 0
\(679\) −8.97056 −0.344259
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 33.6569 1.28784 0.643922 0.765091i \(-0.277306\pi\)
0.643922 + 0.765091i \(0.277306\pi\)
\(684\) 0 0
\(685\) −17.3137 −0.661523
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 13.3137 0.507212
\(690\) 0 0
\(691\) 27.7990 1.05752 0.528762 0.848770i \(-0.322657\pi\)
0.528762 + 0.848770i \(0.322657\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 6.34315 0.240609
\(696\) 0 0
\(697\) −6.34315 −0.240264
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −0.142136 −0.00536839 −0.00268419 0.999996i \(-0.500854\pi\)
−0.00268419 + 0.999996i \(0.500854\pi\)
\(702\) 0 0
\(703\) 32.9706 1.24351
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −45.6569 −1.71710
\(708\) 0 0
\(709\) −7.17157 −0.269334 −0.134667 0.990891i \(-0.542996\pi\)
−0.134667 + 0.990891i \(0.542996\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 33.9411 1.27111
\(714\) 0 0
\(715\) 5.65685 0.211554
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −29.6569 −1.10601 −0.553007 0.833177i \(-0.686520\pi\)
−0.553007 + 0.833177i \(0.686520\pi\)
\(720\) 0 0
\(721\) 4.68629 0.174527
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 8.82843 0.327880
\(726\) 0 0
\(727\) 45.9411 1.70386 0.851931 0.523654i \(-0.175432\pi\)
0.851931 + 0.523654i \(0.175432\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 8.00000 0.295891
\(732\) 0 0
\(733\) −0.343146 −0.0126744 −0.00633719 0.999980i \(-0.502017\pi\)
−0.00633719 + 0.999980i \(0.502017\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −32.0000 −1.17874
\(738\) 0 0
\(739\) 14.1421 0.520227 0.260113 0.965578i \(-0.416240\pi\)
0.260113 + 0.965578i \(0.416240\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −36.2843 −1.33114 −0.665570 0.746335i \(-0.731812\pi\)
−0.665570 + 0.746335i \(0.731812\pi\)
\(744\) 0 0
\(745\) 3.65685 0.133977
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −11.3137 −0.413394
\(750\) 0 0
\(751\) −11.3137 −0.412843 −0.206422 0.978463i \(-0.566182\pi\)
−0.206422 + 0.978463i \(0.566182\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 12.0000 0.436725
\(756\) 0 0
\(757\) 19.9411 0.724773 0.362386 0.932028i \(-0.381962\pi\)
0.362386 + 0.932028i \(0.381962\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −27.6569 −1.00256 −0.501280 0.865285i \(-0.667137\pi\)
−0.501280 + 0.865285i \(0.667137\pi\)
\(762\) 0 0
\(763\) 24.9706 0.903995
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 2.34315 0.0846061
\(768\) 0 0
\(769\) −14.0000 −0.504853 −0.252426 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 53.3137 1.91756 0.958780 0.284148i \(-0.0917107\pi\)
0.958780 + 0.284148i \(0.0917107\pi\)
\(774\) 0 0
\(775\) −4.00000 −0.143684
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −21.6569 −0.775937
\(780\) 0 0
\(781\) −32.0000 −1.14505
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −5.31371 −0.189654
\(786\) 0 0
\(787\) 24.0000 0.855508 0.427754 0.903895i \(-0.359305\pi\)
0.427754 + 0.903895i \(0.359305\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −18.3431 −0.652207
\(792\) 0 0
\(793\) −6.00000 −0.213066
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −16.6274 −0.588973 −0.294487 0.955656i \(-0.595149\pi\)
−0.294487 + 0.955656i \(0.595149\pi\)
\(798\) 0 0
\(799\) 6.62742 0.234461
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −81.9411 −2.89164
\(804\) 0 0
\(805\) 24.0000 0.845889
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −13.3137 −0.468085 −0.234043 0.972226i \(-0.575196\pi\)
−0.234043 + 0.972226i \(0.575196\pi\)
\(810\) 0 0
\(811\) 1.85786 0.0652384 0.0326192 0.999468i \(-0.489615\pi\)
0.0326192 + 0.999468i \(0.489615\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 11.3137 0.396302
\(816\) 0 0
\(817\) 27.3137 0.955586
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −34.2843 −1.19653 −0.598265 0.801299i \(-0.704143\pi\)
−0.598265 + 0.801299i \(0.704143\pi\)
\(822\) 0 0
\(823\) 52.9706 1.84644 0.923219 0.384275i \(-0.125548\pi\)
0.923219 + 0.384275i \(0.125548\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 9.65685 0.335802 0.167901 0.985804i \(-0.446301\pi\)
0.167901 + 0.985804i \(0.446301\pi\)
\(828\) 0 0
\(829\) −53.3137 −1.85166 −0.925831 0.377938i \(-0.876633\pi\)
−0.925831 + 0.377938i \(0.876633\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −0.828427 −0.0287033
\(834\) 0 0
\(835\) −8.97056 −0.310439
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 48.9411 1.68763
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −1.00000 −0.0344010
\(846\) 0 0
\(847\) 59.3970 2.04090
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 98.9117 3.39065
\(852\) 0 0
\(853\) 38.2843 1.31083 0.655414 0.755270i \(-0.272494\pi\)
0.655414 + 0.755270i \(0.272494\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 20.8284 0.711486 0.355743 0.934584i \(-0.384228\pi\)
0.355743 + 0.934584i \(0.384228\pi\)
\(858\) 0 0
\(859\) −37.9411 −1.29453 −0.647267 0.762263i \(-0.724088\pi\)
−0.647267 + 0.762263i \(0.724088\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −28.2843 −0.962808 −0.481404 0.876499i \(-0.659873\pi\)
−0.481404 + 0.876499i \(0.659873\pi\)
\(864\) 0 0
\(865\) −9.31371 −0.316676
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −13.2548 −0.449639
\(870\) 0 0
\(871\) 5.65685 0.191675
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −2.82843 −0.0956183
\(876\) 0 0
\(877\) −51.2548 −1.73075 −0.865376 0.501122i \(-0.832921\pi\)
−0.865376 + 0.501122i \(0.832921\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 10.2843 0.346486 0.173243 0.984879i \(-0.444575\pi\)
0.173243 + 0.984879i \(0.444575\pi\)
\(882\) 0 0
\(883\) −31.3137 −1.05379 −0.526895 0.849930i \(-0.676644\pi\)
−0.526895 + 0.849930i \(0.676644\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −40.4853 −1.35936 −0.679681 0.733508i \(-0.737882\pi\)
−0.679681 + 0.733508i \(0.737882\pi\)
\(888\) 0 0
\(889\) −27.3137 −0.916072
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 22.6274 0.757198
\(894\) 0 0
\(895\) −7.51472 −0.251189
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −35.3137 −1.17778
\(900\) 0 0
\(901\) 11.0294 0.367444
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 7.65685 0.254522
\(906\) 0 0
\(907\) −8.28427 −0.275075 −0.137537 0.990497i \(-0.543919\pi\)
−0.137537 + 0.990497i \(0.543919\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 24.9706 0.827312 0.413656 0.910433i \(-0.364252\pi\)
0.413656 + 0.910433i \(0.364252\pi\)
\(912\) 0 0
\(913\) 35.8823 1.18753
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −17.3726 −0.573693
\(918\) 0 0
\(919\) −41.9411 −1.38351 −0.691755 0.722132i \(-0.743162\pi\)
−0.691755 + 0.722132i \(0.743162\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 5.65685 0.186198
\(924\) 0 0
\(925\) −11.6569 −0.383275
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 33.5980 1.10231 0.551157 0.834402i \(-0.314187\pi\)
0.551157 + 0.834402i \(0.314187\pi\)
\(930\) 0 0
\(931\) −2.82843 −0.0926980
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 4.68629 0.153258
\(936\) 0 0
\(937\) 16.6274 0.543194 0.271597 0.962411i \(-0.412448\pi\)
0.271597 + 0.962411i \(0.412448\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −54.9706 −1.79199 −0.895995 0.444065i \(-0.853536\pi\)
−0.895995 + 0.444065i \(0.853536\pi\)
\(942\) 0 0
\(943\) −64.9706 −2.11573
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 30.3431 0.986020 0.493010 0.870024i \(-0.335897\pi\)
0.493010 + 0.870024i \(0.335897\pi\)
\(948\) 0 0
\(949\) 14.4853 0.470212
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 27.8579 0.902405 0.451202 0.892422i \(-0.350995\pi\)
0.451202 + 0.892422i \(0.350995\pi\)
\(954\) 0 0
\(955\) −11.3137 −0.366103
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 48.9706 1.58134
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −2.48528 −0.0800040
\(966\) 0 0
\(967\) 7.51472 0.241657 0.120829 0.992673i \(-0.461445\pi\)
0.120829 + 0.992673i \(0.461445\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 15.5147 0.497891 0.248946 0.968517i \(-0.419916\pi\)
0.248946 + 0.968517i \(0.419916\pi\)
\(972\) 0 0
\(973\) −17.9411 −0.575166
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 8.34315 0.266921 0.133460 0.991054i \(-0.457391\pi\)
0.133460 + 0.991054i \(0.457391\pi\)
\(978\) 0 0
\(979\) 88.5685 2.83066
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −2.34315 −0.0747347 −0.0373674 0.999302i \(-0.511897\pi\)
−0.0373674 + 0.999302i \(0.511897\pi\)
\(984\) 0 0
\(985\) −13.3137 −0.424210
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 81.9411 2.60558
\(990\) 0 0
\(991\) 42.9117 1.36313 0.681567 0.731755i \(-0.261299\pi\)
0.681567 + 0.731755i \(0.261299\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 10.3431 0.327900
\(996\) 0 0
\(997\) 61.3137 1.94182 0.970912 0.239435i \(-0.0769623\pi\)
0.970912 + 0.239435i \(0.0769623\pi\)
\(998\) 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 9360.2.a.ch.1.2 2
3.2 odd 2 3120.2.a.bc.1.2 2
4.3 odd 2 1170.2.a.o.1.1 2
12.11 even 2 390.2.a.h.1.1 2
20.3 even 4 5850.2.e.bk.5149.4 4
20.7 even 4 5850.2.e.bk.5149.1 4
20.19 odd 2 5850.2.a.cl.1.2 2
60.23 odd 4 1950.2.e.o.1249.2 4
60.47 odd 4 1950.2.e.o.1249.3 4
60.59 even 2 1950.2.a.bd.1.2 2
156.47 odd 4 5070.2.b.q.1351.4 4
156.83 odd 4 5070.2.b.q.1351.1 4
156.155 even 2 5070.2.a.bc.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.h.1.1 2 12.11 even 2
1170.2.a.o.1.1 2 4.3 odd 2
1950.2.a.bd.1.2 2 60.59 even 2
1950.2.e.o.1249.2 4 60.23 odd 4
1950.2.e.o.1249.3 4 60.47 odd 4
3120.2.a.bc.1.2 2 3.2 odd 2
5070.2.a.bc.1.2 2 156.155 even 2
5070.2.b.q.1351.1 4 156.83 odd 4
5070.2.b.q.1351.4 4 156.47 odd 4
5850.2.a.cl.1.2 2 20.19 odd 2
5850.2.e.bk.5149.1 4 20.7 even 4
5850.2.e.bk.5149.4 4 20.3 even 4
9360.2.a.ch.1.2 2 1.1 even 1 trivial