Properties

Label 7920.2.a.cg.1.2
Level $7920$
Weight $2$
Character 7920.1
Self dual yes
Analytic conductor $63.242$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7920 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7920.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.2415184009\)
Analytic rank: \(0\)
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 165)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 7920.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{5} +4.82843 q^{7} +O(q^{10})\) \(q+1.00000 q^{5} +4.82843 q^{7} -1.00000 q^{11} +5.65685 q^{13} +6.82843 q^{17} +1.17157 q^{19} -4.00000 q^{23} +1.00000 q^{25} -0.828427 q^{29} +4.82843 q^{35} +0.343146 q^{37} +0.828427 q^{41} +3.17157 q^{43} -4.00000 q^{47} +16.3137 q^{49} +13.3137 q^{53} -1.00000 q^{55} -4.00000 q^{59} -0.343146 q^{61} +5.65685 q^{65} -5.65685 q^{67} +13.6569 q^{71} -11.3137 q^{73} -4.82843 q^{77} +8.48528 q^{79} -10.0000 q^{83} +6.82843 q^{85} +7.65685 q^{89} +27.3137 q^{91} +1.17157 q^{95} +0.343146 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{5} + 4q^{7} + O(q^{10}) \) \( 2q + 2q^{5} + 4q^{7} - 2q^{11} + 8q^{17} + 8q^{19} - 8q^{23} + 2q^{25} + 4q^{29} + 4q^{35} + 12q^{37} - 4q^{41} + 12q^{43} - 8q^{47} + 10q^{49} + 4q^{53} - 2q^{55} - 8q^{59} - 12q^{61} + 16q^{71} - 4q^{77} - 20q^{83} + 8q^{85} + 4q^{89} + 32q^{91} + 8q^{95} + 12q^{97} + O(q^{100}) \)

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
\(7\) 4.82843 1.82497 0.912487 0.409106i \(-0.134159\pi\)
0.912487 + 0.409106i \(0.134159\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 5.65685 1.56893 0.784465 0.620174i \(-0.212938\pi\)
0.784465 + 0.620174i \(0.212938\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.82843 1.65614 0.828068 0.560627i \(-0.189440\pi\)
0.828068 + 0.560627i \(0.189440\pi\)
\(18\) 0 0
\(19\) 1.17157 0.268777 0.134389 0.990929i \(-0.457093\pi\)
0.134389 + 0.990929i \(0.457093\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −0.828427 −0.153835 −0.0769175 0.997037i \(-0.524508\pi\)
−0.0769175 + 0.997037i \(0.524508\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.82843 0.816153
\(36\) 0 0
\(37\) 0.343146 0.0564128 0.0282064 0.999602i \(-0.491020\pi\)
0.0282064 + 0.999602i \(0.491020\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.828427 0.129379 0.0646893 0.997905i \(-0.479394\pi\)
0.0646893 + 0.997905i \(0.479394\pi\)
\(42\) 0 0
\(43\) 3.17157 0.483660 0.241830 0.970319i \(-0.422252\pi\)
0.241830 + 0.970319i \(0.422252\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.00000 −0.583460 −0.291730 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) 0 0
\(49\) 16.3137 2.33053
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 13.3137 1.82878 0.914389 0.404836i \(-0.132671\pi\)
0.914389 + 0.404836i \(0.132671\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −0.343146 −0.0439353 −0.0219677 0.999759i \(-0.506993\pi\)
−0.0219677 + 0.999759i \(0.506993\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 5.65685 0.701646
\(66\) 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
\(71\) 13.6569 1.62077 0.810385 0.585897i \(-0.199258\pi\)
0.810385 + 0.585897i \(0.199258\pi\)
\(72\) 0 0
\(73\) −11.3137 −1.32417 −0.662085 0.749429i \(-0.730328\pi\)
−0.662085 + 0.749429i \(0.730328\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4.82843 −0.550250
\(78\) 0 0
\(79\) 8.48528 0.954669 0.477334 0.878722i \(-0.341603\pi\)
0.477334 + 0.878722i \(0.341603\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −10.0000 −1.09764 −0.548821 0.835940i \(-0.684923\pi\)
−0.548821 + 0.835940i \(0.684923\pi\)
\(84\) 0 0
\(85\) 6.82843 0.740647
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 7.65685 0.811625 0.405812 0.913956i \(-0.366989\pi\)
0.405812 + 0.913956i \(0.366989\pi\)
\(90\) 0 0
\(91\) 27.3137 2.86325
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.17157 0.120201
\(96\) 0 0
\(97\) 0.343146 0.0348412 0.0174206 0.999848i \(-0.494455\pi\)
0.0174206 + 0.999848i \(0.494455\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −4.82843 −0.480446 −0.240223 0.970718i \(-0.577221\pi\)
−0.240223 + 0.970718i \(0.577221\pi\)
\(102\) 0 0
\(103\) −19.3137 −1.90304 −0.951518 0.307593i \(-0.900477\pi\)
−0.951518 + 0.307593i \(0.900477\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −5.31371 −0.513696 −0.256848 0.966452i \(-0.582684\pi\)
−0.256848 + 0.966452i \(0.582684\pi\)
\(108\) 0 0
\(109\) −5.31371 −0.508961 −0.254480 0.967078i \(-0.581904\pi\)
−0.254480 + 0.967078i \(0.581904\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −14.9706 −1.40831 −0.704156 0.710045i \(-0.748674\pi\)
−0.704156 + 0.710045i \(0.748674\pi\)
\(114\) 0 0
\(115\) −4.00000 −0.373002
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 32.9706 3.02241
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −2.48528 −0.220533 −0.110267 0.993902i \(-0.535170\pi\)
−0.110267 + 0.993902i \(0.535170\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −19.3137 −1.68745 −0.843723 0.536778i \(-0.819641\pi\)
−0.843723 + 0.536778i \(0.819641\pi\)
\(132\) 0 0
\(133\) 5.65685 0.490511
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −9.31371 −0.795724 −0.397862 0.917445i \(-0.630248\pi\)
−0.397862 + 0.917445i \(0.630248\pi\)
\(138\) 0 0
\(139\) 16.4853 1.39826 0.699132 0.714993i \(-0.253570\pi\)
0.699132 + 0.714993i \(0.253570\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −5.65685 −0.473050
\(144\) 0 0
\(145\) −0.828427 −0.0687971
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −18.4853 −1.51437 −0.757187 0.653199i \(-0.773427\pi\)
−0.757187 + 0.653199i \(0.773427\pi\)
\(150\) 0 0
\(151\) 0.485281 0.0394916 0.0197458 0.999805i \(-0.493714\pi\)
0.0197458 + 0.999805i \(0.493714\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −19.3137 −1.52213
\(162\) 0 0
\(163\) −15.3137 −1.19946 −0.599731 0.800202i \(-0.704726\pi\)
−0.599731 + 0.800202i \(0.704726\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 9.31371 0.720716 0.360358 0.932814i \(-0.382654\pi\)
0.360358 + 0.932814i \(0.382654\pi\)
\(168\) 0 0
\(169\) 19.0000 1.46154
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 2.82843 0.215041 0.107521 0.994203i \(-0.465709\pi\)
0.107521 + 0.994203i \(0.465709\pi\)
\(174\) 0 0
\(175\) 4.82843 0.364995
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −6.34315 −0.474109 −0.237054 0.971496i \(-0.576182\pi\)
−0.237054 + 0.971496i \(0.576182\pi\)
\(180\) 0 0
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0.343146 0.0252286
\(186\) 0 0
\(187\) −6.82843 −0.499344
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −5.65685 −0.409316 −0.204658 0.978834i \(-0.565608\pi\)
−0.204658 + 0.978834i \(0.565608\pi\)
\(192\) 0 0
\(193\) 2.34315 0.168663 0.0843317 0.996438i \(-0.473124\pi\)
0.0843317 + 0.996438i \(0.473124\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −8.48528 −0.604551 −0.302276 0.953221i \(-0.597746\pi\)
−0.302276 + 0.953221i \(0.597746\pi\)
\(198\) 0 0
\(199\) 10.3431 0.733206 0.366603 0.930377i \(-0.380521\pi\)
0.366603 + 0.930377i \(0.380521\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −4.00000 −0.280745
\(204\) 0 0
\(205\) 0.828427 0.0578599
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −1.17157 −0.0810394
\(210\) 0 0
\(211\) −6.82843 −0.470088 −0.235044 0.971985i \(-0.575523\pi\)
−0.235044 + 0.971985i \(0.575523\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 3.17157 0.216299
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 38.6274 2.59836
\(222\) 0 0
\(223\) 17.6569 1.18239 0.591195 0.806529i \(-0.298656\pi\)
0.591195 + 0.806529i \(0.298656\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −14.0000 −0.929213 −0.464606 0.885517i \(-0.653804\pi\)
−0.464606 + 0.885517i \(0.653804\pi\)
\(228\) 0 0
\(229\) −2.00000 −0.132164 −0.0660819 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 13.1716 0.862898 0.431449 0.902137i \(-0.358002\pi\)
0.431449 + 0.902137i \(0.358002\pi\)
\(234\) 0 0
\(235\) −4.00000 −0.260931
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 6.34315 0.410304 0.205152 0.978730i \(-0.434231\pi\)
0.205152 + 0.978730i \(0.434231\pi\)
\(240\) 0 0
\(241\) −23.6569 −1.52387 −0.761936 0.647652i \(-0.775751\pi\)
−0.761936 + 0.647652i \(0.775751\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 16.3137 1.04224
\(246\) 0 0
\(247\) 6.62742 0.421692
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −12.9706 −0.818695 −0.409347 0.912379i \(-0.634244\pi\)
−0.409347 + 0.912379i \(0.634244\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 27.6569 1.72519 0.862594 0.505898i \(-0.168839\pi\)
0.862594 + 0.505898i \(0.168839\pi\)
\(258\) 0 0
\(259\) 1.65685 0.102952
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 18.0000 1.10993 0.554964 0.831875i \(-0.312732\pi\)
0.554964 + 0.831875i \(0.312732\pi\)
\(264\) 0 0
\(265\) 13.3137 0.817855
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 24.6274 1.50156 0.750780 0.660552i \(-0.229678\pi\)
0.750780 + 0.660552i \(0.229678\pi\)
\(270\) 0 0
\(271\) −27.7990 −1.68867 −0.844334 0.535817i \(-0.820004\pi\)
−0.844334 + 0.535817i \(0.820004\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) −13.6569 −0.820561 −0.410280 0.911959i \(-0.634569\pi\)
−0.410280 + 0.911959i \(0.634569\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 16.8284 1.00390 0.501950 0.864897i \(-0.332616\pi\)
0.501950 + 0.864897i \(0.332616\pi\)
\(282\) 0 0
\(283\) 3.17157 0.188530 0.0942652 0.995547i \(-0.469950\pi\)
0.0942652 + 0.995547i \(0.469950\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 4.00000 0.236113
\(288\) 0 0
\(289\) 29.6274 1.74279
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1.17157 0.0684440 0.0342220 0.999414i \(-0.489105\pi\)
0.0342220 + 0.999414i \(0.489105\pi\)
\(294\) 0 0
\(295\) −4.00000 −0.232889
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −22.6274 −1.30858
\(300\) 0 0
\(301\) 15.3137 0.882667
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −0.343146 −0.0196485
\(306\) 0 0
\(307\) 8.82843 0.503865 0.251932 0.967745i \(-0.418934\pi\)
0.251932 + 0.967745i \(0.418934\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 19.3137 1.09518 0.547590 0.836747i \(-0.315545\pi\)
0.547590 + 0.836747i \(0.315545\pi\)
\(312\) 0 0
\(313\) 4.34315 0.245489 0.122745 0.992438i \(-0.460830\pi\)
0.122745 + 0.992438i \(0.460830\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −30.2843 −1.70093 −0.850467 0.526028i \(-0.823681\pi\)
−0.850467 + 0.526028i \(0.823681\pi\)
\(318\) 0 0
\(319\) 0.828427 0.0463830
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 8.00000 0.445132
\(324\) 0 0
\(325\) 5.65685 0.313786
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −19.3137 −1.06480
\(330\) 0 0
\(331\) −17.6569 −0.970508 −0.485254 0.874373i \(-0.661273\pi\)
−0.485254 + 0.874373i \(0.661273\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −5.65685 −0.309067
\(336\) 0 0
\(337\) −19.3137 −1.05208 −0.526042 0.850458i \(-0.676325\pi\)
−0.526042 + 0.850458i \(0.676325\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 44.9706 2.42818
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −6.68629 −0.358939 −0.179469 0.983764i \(-0.557438\pi\)
−0.179469 + 0.983764i \(0.557438\pi\)
\(348\) 0 0
\(349\) −22.9706 −1.22959 −0.614793 0.788688i \(-0.710760\pi\)
−0.614793 + 0.788688i \(0.710760\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −26.0000 −1.38384 −0.691920 0.721974i \(-0.743235\pi\)
−0.691920 + 0.721974i \(0.743235\pi\)
\(354\) 0 0
\(355\) 13.6569 0.724831
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 12.0000 0.633336 0.316668 0.948536i \(-0.397436\pi\)
0.316668 + 0.948536i \(0.397436\pi\)
\(360\) 0 0
\(361\) −17.6274 −0.927759
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −11.3137 −0.592187
\(366\) 0 0
\(367\) 1.65685 0.0864871 0.0432435 0.999065i \(-0.486231\pi\)
0.0432435 + 0.999065i \(0.486231\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 64.2843 3.33747
\(372\) 0 0
\(373\) −34.6274 −1.79294 −0.896470 0.443105i \(-0.853877\pi\)
−0.896470 + 0.443105i \(0.853877\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −4.68629 −0.241356
\(378\) 0 0
\(379\) 0.686292 0.0352524 0.0176262 0.999845i \(-0.494389\pi\)
0.0176262 + 0.999845i \(0.494389\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −8.00000 −0.408781 −0.204390 0.978889i \(-0.565521\pi\)
−0.204390 + 0.978889i \(0.565521\pi\)
\(384\) 0 0
\(385\) −4.82843 −0.246079
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 12.3431 0.625822 0.312911 0.949782i \(-0.398696\pi\)
0.312911 + 0.949782i \(0.398696\pi\)
\(390\) 0 0
\(391\) −27.3137 −1.38131
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 8.48528 0.426941
\(396\) 0 0
\(397\) 18.9706 0.952105 0.476053 0.879417i \(-0.342067\pi\)
0.476053 + 0.879417i \(0.342067\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 29.3137 1.46386 0.731928 0.681382i \(-0.238621\pi\)
0.731928 + 0.681382i \(0.238621\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −0.343146 −0.0170091
\(408\) 0 0
\(409\) −8.34315 −0.412542 −0.206271 0.978495i \(-0.566133\pi\)
−0.206271 + 0.978495i \(0.566133\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −19.3137 −0.950365
\(414\) 0 0
\(415\) −10.0000 −0.490881
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −3.02944 −0.147998 −0.0739988 0.997258i \(-0.523576\pi\)
−0.0739988 + 0.997258i \(0.523576\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 6.82843 0.331227
\(426\) 0 0
\(427\) −1.65685 −0.0801808
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 10.3431 0.498212 0.249106 0.968476i \(-0.419863\pi\)
0.249106 + 0.968476i \(0.419863\pi\)
\(432\) 0 0
\(433\) −4.34315 −0.208718 −0.104359 0.994540i \(-0.533279\pi\)
−0.104359 + 0.994540i \(0.533279\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −4.68629 −0.224176
\(438\) 0 0
\(439\) 3.51472 0.167748 0.0838742 0.996476i \(-0.473271\pi\)
0.0838742 + 0.996476i \(0.473271\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 0 0
\(445\) 7.65685 0.362970
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 2.97056 0.140190 0.0700948 0.997540i \(-0.477670\pi\)
0.0700948 + 0.997540i \(0.477670\pi\)
\(450\) 0 0
\(451\) −0.828427 −0.0390091
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 27.3137 1.28049
\(456\) 0 0
\(457\) −0.686292 −0.0321034 −0.0160517 0.999871i \(-0.505110\pi\)
−0.0160517 + 0.999871i \(0.505110\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 28.1421 1.31071 0.655355 0.755321i \(-0.272519\pi\)
0.655355 + 0.755321i \(0.272519\pi\)
\(462\) 0 0
\(463\) 28.9706 1.34638 0.673188 0.739471i \(-0.264924\pi\)
0.673188 + 0.739471i \(0.264924\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 22.6274 1.04707 0.523536 0.852004i \(-0.324613\pi\)
0.523536 + 0.852004i \(0.324613\pi\)
\(468\) 0 0
\(469\) −27.3137 −1.26123
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −3.17157 −0.145829
\(474\) 0 0
\(475\) 1.17157 0.0537555
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 3.02944 0.138419 0.0692093 0.997602i \(-0.477952\pi\)
0.0692093 + 0.997602i \(0.477952\pi\)
\(480\) 0 0
\(481\) 1.94113 0.0885077
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0.343146 0.0155814
\(486\) 0 0
\(487\) −20.9706 −0.950267 −0.475133 0.879914i \(-0.657600\pi\)
−0.475133 + 0.879914i \(0.657600\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 25.6569 1.15788 0.578939 0.815371i \(-0.303467\pi\)
0.578939 + 0.815371i \(0.303467\pi\)
\(492\) 0 0
\(493\) −5.65685 −0.254772
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 65.9411 2.95786
\(498\) 0 0
\(499\) 33.6569 1.50669 0.753344 0.657627i \(-0.228440\pi\)
0.753344 + 0.657627i \(0.228440\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −5.31371 −0.236927 −0.118463 0.992958i \(-0.537797\pi\)
−0.118463 + 0.992958i \(0.537797\pi\)
\(504\) 0 0
\(505\) −4.82843 −0.214862
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −41.3137 −1.83120 −0.915599 0.402093i \(-0.868283\pi\)
−0.915599 + 0.402093i \(0.868283\pi\)
\(510\) 0 0
\(511\) −54.6274 −2.41657
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −19.3137 −0.851064
\(516\) 0 0
\(517\) 4.00000 0.175920
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −12.6274 −0.553217 −0.276609 0.960983i \(-0.589211\pi\)
−0.276609 + 0.960983i \(0.589211\pi\)
\(522\) 0 0
\(523\) 26.4853 1.15812 0.579060 0.815285i \(-0.303420\pi\)
0.579060 + 0.815285i \(0.303420\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 4.68629 0.202986
\(534\) 0 0
\(535\) −5.31371 −0.229732
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −16.3137 −0.702681
\(540\) 0 0
\(541\) −5.31371 −0.228454 −0.114227 0.993455i \(-0.536439\pi\)
−0.114227 + 0.993455i \(0.536439\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −5.31371 −0.227614
\(546\) 0 0
\(547\) −20.1421 −0.861216 −0.430608 0.902539i \(-0.641701\pi\)
−0.430608 + 0.902539i \(0.641701\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −0.970563 −0.0413474
\(552\) 0 0
\(553\) 40.9706 1.74225
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −10.8284 −0.458815 −0.229408 0.973330i \(-0.573679\pi\)
−0.229408 + 0.973330i \(0.573679\pi\)
\(558\) 0 0
\(559\) 17.9411 0.758829
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 20.3431 0.857361 0.428681 0.903456i \(-0.358979\pi\)
0.428681 + 0.903456i \(0.358979\pi\)
\(564\) 0 0
\(565\) −14.9706 −0.629816
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −15.4558 −0.647943 −0.323971 0.946067i \(-0.605018\pi\)
−0.323971 + 0.946067i \(0.605018\pi\)
\(570\) 0 0
\(571\) −0.485281 −0.0203084 −0.0101542 0.999948i \(-0.503232\pi\)
−0.0101542 + 0.999948i \(0.503232\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −4.00000 −0.166812
\(576\) 0 0
\(577\) 14.0000 0.582828 0.291414 0.956597i \(-0.405874\pi\)
0.291414 + 0.956597i \(0.405874\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −48.2843 −2.00317
\(582\) 0 0
\(583\) −13.3137 −0.551397
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −30.6274 −1.26413 −0.632064 0.774916i \(-0.717792\pi\)
−0.632064 + 0.774916i \(0.717792\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 17.1716 0.705152 0.352576 0.935783i \(-0.385306\pi\)
0.352576 + 0.935783i \(0.385306\pi\)
\(594\) 0 0
\(595\) 32.9706 1.35166
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 4.68629 0.191477 0.0957383 0.995407i \(-0.469479\pi\)
0.0957383 + 0.995407i \(0.469479\pi\)
\(600\) 0 0
\(601\) 17.3137 0.706241 0.353120 0.935578i \(-0.385121\pi\)
0.353120 + 0.935578i \(0.385121\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.00000 0.0406558
\(606\) 0 0
\(607\) −18.4853 −0.750294 −0.375147 0.926965i \(-0.622408\pi\)
−0.375147 + 0.926965i \(0.622408\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −22.6274 −0.915407
\(612\) 0 0
\(613\) 21.9411 0.886194 0.443097 0.896474i \(-0.353880\pi\)
0.443097 + 0.896474i \(0.353880\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −11.6569 −0.469287 −0.234644 0.972081i \(-0.575392\pi\)
−0.234644 + 0.972081i \(0.575392\pi\)
\(618\) 0 0
\(619\) 25.6569 1.03124 0.515618 0.856819i \(-0.327562\pi\)
0.515618 + 0.856819i \(0.327562\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 36.9706 1.48119
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 2.34315 0.0934273
\(630\) 0 0
\(631\) −34.3431 −1.36718 −0.683590 0.729867i \(-0.739582\pi\)
−0.683590 + 0.729867i \(0.739582\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −2.48528 −0.0986254
\(636\) 0 0
\(637\) 92.2843 3.65644
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 26.9706 1.06527 0.532637 0.846344i \(-0.321201\pi\)
0.532637 + 0.846344i \(0.321201\pi\)
\(642\) 0 0
\(643\) 29.9411 1.18076 0.590381 0.807124i \(-0.298977\pi\)
0.590381 + 0.807124i \(0.298977\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 27.3137 1.07381 0.536906 0.843642i \(-0.319593\pi\)
0.536906 + 0.843642i \(0.319593\pi\)
\(648\) 0 0
\(649\) 4.00000 0.157014
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 26.9706 1.05544 0.527720 0.849418i \(-0.323047\pi\)
0.527720 + 0.849418i \(0.323047\pi\)
\(654\) 0 0
\(655\) −19.3137 −0.754649
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 7.31371 0.284902 0.142451 0.989802i \(-0.454502\pi\)
0.142451 + 0.989802i \(0.454502\pi\)
\(660\) 0 0
\(661\) −13.3137 −0.517843 −0.258922 0.965898i \(-0.583367\pi\)
−0.258922 + 0.965898i \(0.583367\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 5.65685 0.219363
\(666\) 0 0
\(667\) 3.31371 0.128307
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0.343146 0.0132470
\(672\) 0 0
\(673\) 29.6569 1.14319 0.571594 0.820537i \(-0.306325\pi\)
0.571594 + 0.820537i \(0.306325\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 21.4558 0.824615 0.412308 0.911045i \(-0.364723\pi\)
0.412308 + 0.911045i \(0.364723\pi\)
\(678\) 0 0
\(679\) 1.65685 0.0635842
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 24.0000 0.918334 0.459167 0.888350i \(-0.348148\pi\)
0.459167 + 0.888350i \(0.348148\pi\)
\(684\) 0 0
\(685\) −9.31371 −0.355859
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 75.3137 2.86922
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 16.4853 0.625322
\(696\) 0 0
\(697\) 5.65685 0.214269
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −7.85786 −0.296787 −0.148394 0.988928i \(-0.547410\pi\)
−0.148394 + 0.988928i \(0.547410\pi\)
\(702\) 0 0
\(703\) 0.402020 0.0151625
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −23.3137 −0.876802
\(708\) 0 0
\(709\) 29.3137 1.10090 0.550450 0.834868i \(-0.314456\pi\)
0.550450 + 0.834868i \(0.314456\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −5.65685 −0.211554
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 31.5980 1.17841 0.589203 0.807985i \(-0.299442\pi\)
0.589203 + 0.807985i \(0.299442\pi\)
\(720\) 0 0
\(721\) −93.2548 −3.47299
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −0.828427 −0.0307670
\(726\) 0 0
\(727\) 33.9411 1.25881 0.629403 0.777079i \(-0.283299\pi\)
0.629403 + 0.777079i \(0.283299\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 21.6569 0.801008
\(732\) 0 0
\(733\) −17.6569 −0.652171 −0.326085 0.945340i \(-0.605730\pi\)
−0.326085 + 0.945340i \(0.605730\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 5.65685 0.208373
\(738\) 0 0
\(739\) −47.1127 −1.73307 −0.866534 0.499118i \(-0.833658\pi\)
−0.866534 + 0.499118i \(0.833658\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 47.6569 1.74836 0.874180 0.485602i \(-0.161399\pi\)
0.874180 + 0.485602i \(0.161399\pi\)
\(744\) 0 0
\(745\) −18.4853 −0.677248
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −25.6569 −0.937481
\(750\) 0 0
\(751\) 36.2843 1.32403 0.662016 0.749490i \(-0.269701\pi\)
0.662016 + 0.749490i \(0.269701\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0.485281 0.0176612
\(756\) 0 0
\(757\) 8.62742 0.313569 0.156784 0.987633i \(-0.449887\pi\)
0.156784 + 0.987633i \(0.449887\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 23.1716 0.839969 0.419984 0.907531i \(-0.362036\pi\)
0.419984 + 0.907531i \(0.362036\pi\)
\(762\) 0 0
\(763\) −25.6569 −0.928840
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −22.6274 −0.817029
\(768\) 0 0
\(769\) 33.3137 1.20132 0.600662 0.799503i \(-0.294904\pi\)
0.600662 + 0.799503i \(0.294904\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 7.65685 0.275398 0.137699 0.990474i \(-0.456029\pi\)
0.137699 + 0.990474i \(0.456029\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0.970563 0.0347740
\(780\) 0 0
\(781\) −13.6569 −0.488681
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 18.0000 0.642448
\(786\) 0 0
\(787\) −8.14214 −0.290236 −0.145118 0.989414i \(-0.546356\pi\)
−0.145118 + 0.989414i \(0.546356\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −72.2843 −2.57013
\(792\) 0 0
\(793\) −1.94113 −0.0689314
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.02944 −0.0364645 −0.0182323 0.999834i \(-0.505804\pi\)
−0.0182323 + 0.999834i \(0.505804\pi\)
\(798\) 0 0
\(799\) −27.3137 −0.966290
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 11.3137 0.399252
\(804\) 0 0
\(805\) −19.3137 −0.680719
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 56.4264 1.98385 0.991923 0.126838i \(-0.0404829\pi\)
0.991923 + 0.126838i \(0.0404829\pi\)
\(810\) 0 0
\(811\) 16.4853 0.578877 0.289438 0.957197i \(-0.406532\pi\)
0.289438 + 0.957197i \(0.406532\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −15.3137 −0.536416
\(816\) 0 0
\(817\) 3.71573 0.129997
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 7.17157 0.250290 0.125145 0.992138i \(-0.460060\pi\)
0.125145 + 0.992138i \(0.460060\pi\)
\(822\) 0 0
\(823\) −16.0000 −0.557725 −0.278862 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 18.6863 0.649786 0.324893 0.945751i \(-0.394672\pi\)
0.324893 + 0.945751i \(0.394672\pi\)
\(828\) 0 0
\(829\) −38.0000 −1.31979 −0.659897 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 111.397 3.85968
\(834\) 0 0
\(835\) 9.31371 0.322314
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 22.6274 0.781185 0.390593 0.920564i \(-0.372270\pi\)
0.390593 + 0.920564i \(0.372270\pi\)
\(840\) 0 0
\(841\) −28.3137 −0.976335
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 19.0000 0.653620
\(846\) 0 0
\(847\) 4.82843 0.165907
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −1.37258 −0.0470515
\(852\) 0 0
\(853\) −31.3137 −1.07216 −0.536080 0.844167i \(-0.680096\pi\)
−0.536080 + 0.844167i \(0.680096\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 11.5147 0.393335 0.196668 0.980470i \(-0.436988\pi\)
0.196668 + 0.980470i \(0.436988\pi\)
\(858\) 0 0
\(859\) 19.0294 0.649276 0.324638 0.945838i \(-0.394758\pi\)
0.324638 + 0.945838i \(0.394758\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −43.3137 −1.47442 −0.737208 0.675666i \(-0.763856\pi\)
−0.737208 + 0.675666i \(0.763856\pi\)
\(864\) 0 0
\(865\) 2.82843 0.0961694
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −8.48528 −0.287843
\(870\) 0 0
\(871\) −32.0000 −1.08428
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 4.82843 0.163231
\(876\) 0 0
\(877\) −42.6274 −1.43943 −0.719713 0.694272i \(-0.755727\pi\)
−0.719713 + 0.694272i \(0.755727\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 13.0294 0.438973 0.219486 0.975616i \(-0.429562\pi\)
0.219486 + 0.975616i \(0.429562\pi\)
\(882\) 0 0
\(883\) 50.6274 1.70375 0.851874 0.523747i \(-0.175466\pi\)
0.851874 + 0.523747i \(0.175466\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −4.34315 −0.145829 −0.0729143 0.997338i \(-0.523230\pi\)
−0.0729143 + 0.997338i \(0.523230\pi\)
\(888\) 0 0
\(889\) −12.0000 −0.402467
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −4.68629 −0.156821
\(894\) 0 0
\(895\) −6.34315 −0.212028
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 90.9117 3.02871
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −14.0000 −0.465376
\(906\) 0 0
\(907\) −7.02944 −0.233409 −0.116704 0.993167i \(-0.537233\pi\)
−0.116704 + 0.993167i \(0.537233\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −15.0294 −0.497947 −0.248974 0.968510i \(-0.580093\pi\)
−0.248974 + 0.968510i \(0.580093\pi\)
\(912\) 0 0
\(913\) 10.0000 0.330952
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −93.2548 −3.07955
\(918\) 0 0
\(919\) −28.4853 −0.939643 −0.469821 0.882762i \(-0.655682\pi\)
−0.469821 + 0.882762i \(0.655682\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 77.2548 2.54287
\(924\) 0 0
\(925\) 0.343146 0.0112826
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −33.5980 −1.10231 −0.551157 0.834402i \(-0.685813\pi\)
−0.551157 + 0.834402i \(0.685813\pi\)
\(930\) 0 0
\(931\) 19.1127 0.626393
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −6.82843 −0.223313
\(936\) 0 0
\(937\) 44.9706 1.46912 0.734562 0.678541i \(-0.237388\pi\)
0.734562 + 0.678541i \(0.237388\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −38.7696 −1.26385 −0.631926 0.775029i \(-0.717735\pi\)
−0.631926 + 0.775029i \(0.717735\pi\)
\(942\) 0 0
\(943\) −3.31371 −0.107909
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −38.6274 −1.25522 −0.627611 0.778527i \(-0.715967\pi\)
−0.627611 + 0.778527i \(0.715967\pi\)
\(948\) 0 0
\(949\) −64.0000 −2.07753
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −27.7990 −0.900498 −0.450249 0.892903i \(-0.648665\pi\)
−0.450249 + 0.892903i \(0.648665\pi\)
\(954\) 0 0
\(955\) −5.65685 −0.183052
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −44.9706 −1.45218
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 2.34315 0.0754285
\(966\) 0 0
\(967\) 39.4558 1.26881 0.634407 0.772999i \(-0.281244\pi\)
0.634407 + 0.772999i \(0.281244\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 10.6274 0.341050 0.170525 0.985353i \(-0.445454\pi\)
0.170525 + 0.985353i \(0.445454\pi\)
\(972\) 0 0
\(973\) 79.5980 2.55179
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −25.3137 −0.809857 −0.404929 0.914348i \(-0.632704\pi\)
−0.404929 + 0.914348i \(0.632704\pi\)
\(978\) 0 0
\(979\) −7.65685 −0.244714
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 14.6274 0.466542 0.233271 0.972412i \(-0.425057\pi\)
0.233271 + 0.972412i \(0.425057\pi\)
\(984\) 0 0
\(985\) −8.48528 −0.270364
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −12.6863 −0.403401
\(990\) 0 0
\(991\) −14.6274 −0.464655 −0.232328 0.972638i \(-0.574634\pi\)
−0.232328 + 0.972638i \(0.574634\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 10.3431 0.327900
\(996\) 0 0
\(997\) −16.6863 −0.528460 −0.264230 0.964460i \(-0.585118\pi\)
−0.264230 + 0.964460i \(0.585118\pi\)
\(998\) 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 7920.2.a.cg.1.2 2
3.2 odd 2 2640.2.a.bb.1.2 2
4.3 odd 2 495.2.a.d.1.1 2
12.11 even 2 165.2.a.a.1.2 2
20.3 even 4 2475.2.c.m.199.3 4
20.7 even 4 2475.2.c.m.199.2 4
20.19 odd 2 2475.2.a.m.1.2 2
44.43 even 2 5445.2.a.m.1.2 2
60.23 odd 4 825.2.c.e.199.2 4
60.47 odd 4 825.2.c.e.199.3 4
60.59 even 2 825.2.a.g.1.1 2
84.83 odd 2 8085.2.a.ba.1.2 2
132.131 odd 2 1815.2.a.k.1.1 2
660.659 odd 2 9075.2.a.v.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.a.1.2 2 12.11 even 2
495.2.a.d.1.1 2 4.3 odd 2
825.2.a.g.1.1 2 60.59 even 2
825.2.c.e.199.2 4 60.23 odd 4
825.2.c.e.199.3 4 60.47 odd 4
1815.2.a.k.1.1 2 132.131 odd 2
2475.2.a.m.1.2 2 20.19 odd 2
2475.2.c.m.199.2 4 20.7 even 4
2475.2.c.m.199.3 4 20.3 even 4
2640.2.a.bb.1.2 2 3.2 odd 2
5445.2.a.m.1.2 2 44.43 even 2
7920.2.a.cg.1.2 2 1.1 even 1 trivial
8085.2.a.ba.1.2 2 84.83 odd 2
9075.2.a.v.1.2 2 660.659 odd 2