Properties

Label 4680.2.a.bf.1.2
Level $4680$
Weight $2$
Character 4680.1
Self dual yes
Analytic conductor $37.370$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4680.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(37.3699881460\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1560)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.37228\) of defining polynomial
Character \(\chi\) \(=\) 4680.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{5} +3.37228 q^{7} +O(q^{10})\) \(q+1.00000 q^{5} +3.37228 q^{7} +3.37228 q^{11} -1.00000 q^{13} +1.37228 q^{17} -6.74456 q^{19} +0.627719 q^{23} +1.00000 q^{25} +2.00000 q^{29} +6.74456 q^{31} +3.37228 q^{35} +5.37228 q^{37} +1.37228 q^{41} -4.00000 q^{43} +1.25544 q^{47} +4.37228 q^{49} +9.37228 q^{53} +3.37228 q^{55} +8.00000 q^{59} +8.11684 q^{61} -1.00000 q^{65} -4.00000 q^{67} +11.3723 q^{71} -15.4891 q^{73} +11.3723 q^{77} -16.8614 q^{79} +12.0000 q^{83} +1.37228 q^{85} -13.3723 q^{89} -3.37228 q^{91} -6.74456 q^{95} +2.62772 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} + q^{7} + q^{11} - 2 q^{13} - 3 q^{17} - 2 q^{19} + 7 q^{23} + 2 q^{25} + 4 q^{29} + 2 q^{31} + q^{35} + 5 q^{37} - 3 q^{41} - 8 q^{43} + 14 q^{47} + 3 q^{49} + 13 q^{53} + q^{55} + 16 q^{59} - q^{61} - 2 q^{65} - 8 q^{67} + 17 q^{71} - 8 q^{73} + 17 q^{77} - 5 q^{79} + 24 q^{83} - 3 q^{85} - 21 q^{89} - q^{91} - 2 q^{95} + 11 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(7\) 3.37228 1.27460 0.637301 0.770615i \(-0.280051\pi\)
0.637301 + 0.770615i \(0.280051\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.37228 1.01678 0.508391 0.861127i \(-0.330241\pi\)
0.508391 + 0.861127i \(0.330241\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.37228 0.332827 0.166414 0.986056i \(-0.446781\pi\)
0.166414 + 0.986056i \(0.446781\pi\)
\(18\) 0 0
\(19\) −6.74456 −1.54731 −0.773654 0.633608i \(-0.781573\pi\)
−0.773654 + 0.633608i \(0.781573\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.627719 0.130888 0.0654442 0.997856i \(-0.479154\pi\)
0.0654442 + 0.997856i \(0.479154\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 6.74456 1.21136 0.605680 0.795709i \(-0.292901\pi\)
0.605680 + 0.795709i \(0.292901\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.37228 0.570020
\(36\) 0 0
\(37\) 5.37228 0.883198 0.441599 0.897213i \(-0.354411\pi\)
0.441599 + 0.897213i \(0.354411\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.37228 0.214314 0.107157 0.994242i \(-0.465825\pi\)
0.107157 + 0.994242i \(0.465825\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.25544 0.183124 0.0915622 0.995799i \(-0.470814\pi\)
0.0915622 + 0.995799i \(0.470814\pi\)
\(48\) 0 0
\(49\) 4.37228 0.624612
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 9.37228 1.28738 0.643691 0.765286i \(-0.277402\pi\)
0.643691 + 0.765286i \(0.277402\pi\)
\(54\) 0 0
\(55\) 3.37228 0.454718
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) 8.11684 1.03926 0.519628 0.854393i \(-0.326071\pi\)
0.519628 + 0.854393i \(0.326071\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 11.3723 1.34964 0.674821 0.737982i \(-0.264221\pi\)
0.674821 + 0.737982i \(0.264221\pi\)
\(72\) 0 0
\(73\) −15.4891 −1.81286 −0.906432 0.422351i \(-0.861205\pi\)
−0.906432 + 0.422351i \(0.861205\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 11.3723 1.29599
\(78\) 0 0
\(79\) −16.8614 −1.89706 −0.948528 0.316693i \(-0.897428\pi\)
−0.948528 + 0.316693i \(0.897428\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) 1.37228 0.148845
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −13.3723 −1.41746 −0.708729 0.705480i \(-0.750731\pi\)
−0.708729 + 0.705480i \(0.750731\pi\)
\(90\) 0 0
\(91\) −3.37228 −0.353511
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −6.74456 −0.691978
\(96\) 0 0
\(97\) 2.62772 0.266804 0.133402 0.991062i \(-0.457410\pi\)
0.133402 + 0.991062i \(0.457410\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −4.74456 −0.472102 −0.236051 0.971741i \(-0.575853\pi\)
−0.236051 + 0.971741i \(0.575853\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 8.62772 0.834073 0.417037 0.908890i \(-0.363069\pi\)
0.417037 + 0.908890i \(0.363069\pi\)
\(108\) 0 0
\(109\) 8.74456 0.837577 0.418789 0.908084i \(-0.362455\pi\)
0.418789 + 0.908084i \(0.362455\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 0.627719 0.0585351
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 4.62772 0.424222
\(120\) 0 0
\(121\) 0.372281 0.0338438
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −16.2337 −1.44051 −0.720253 0.693711i \(-0.755974\pi\)
−0.720253 + 0.693711i \(0.755974\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 18.7446 1.63772 0.818860 0.573993i \(-0.194606\pi\)
0.818860 + 0.573993i \(0.194606\pi\)
\(132\) 0 0
\(133\) −22.7446 −1.97220
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −11.4891 −0.981582 −0.490791 0.871277i \(-0.663292\pi\)
−0.490791 + 0.871277i \(0.663292\pi\)
\(138\) 0 0
\(139\) 7.37228 0.625309 0.312654 0.949867i \(-0.398782\pi\)
0.312654 + 0.949867i \(0.398782\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −3.37228 −0.282004
\(144\) 0 0
\(145\) 2.00000 0.166091
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −13.3723 −1.09550 −0.547750 0.836642i \(-0.684515\pi\)
−0.547750 + 0.836642i \(0.684515\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 6.74456 0.541736
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 2.11684 0.166831
\(162\) 0 0
\(163\) −0.627719 −0.0491667 −0.0245834 0.999698i \(-0.507826\pi\)
−0.0245834 + 0.999698i \(0.507826\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −7.48913 −0.569388 −0.284694 0.958618i \(-0.591892\pi\)
−0.284694 + 0.958618i \(0.591892\pi\)
\(174\) 0 0
\(175\) 3.37228 0.254921
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −24.2337 −1.81131 −0.905655 0.424014i \(-0.860621\pi\)
−0.905655 + 0.424014i \(0.860621\pi\)
\(180\) 0 0
\(181\) −5.37228 −0.399319 −0.199659 0.979865i \(-0.563984\pi\)
−0.199659 + 0.979865i \(0.563984\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 5.37228 0.394978
\(186\) 0 0
\(187\) 4.62772 0.338412
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 0 0
\(193\) 9.37228 0.674632 0.337316 0.941392i \(-0.390481\pi\)
0.337316 + 0.941392i \(0.390481\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 10.2337 0.729120 0.364560 0.931180i \(-0.381219\pi\)
0.364560 + 0.931180i \(0.381219\pi\)
\(198\) 0 0
\(199\) 21.4891 1.52332 0.761662 0.647975i \(-0.224384\pi\)
0.761662 + 0.647975i \(0.224384\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 6.74456 0.473375
\(204\) 0 0
\(205\) 1.37228 0.0958443
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −22.7446 −1.57327
\(210\) 0 0
\(211\) 22.9783 1.58189 0.790944 0.611889i \(-0.209590\pi\)
0.790944 + 0.611889i \(0.209590\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −4.00000 −0.272798
\(216\) 0 0
\(217\) 22.7446 1.54400
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −1.37228 −0.0923096
\(222\) 0 0
\(223\) 18.9783 1.27088 0.635439 0.772151i \(-0.280819\pi\)
0.635439 + 0.772151i \(0.280819\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 18.7446 1.24412 0.622060 0.782969i \(-0.286296\pi\)
0.622060 + 0.782969i \(0.286296\pi\)
\(228\) 0 0
\(229\) 2.00000 0.132164 0.0660819 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 18.6277 1.22034 0.610171 0.792270i \(-0.291101\pi\)
0.610171 + 0.792270i \(0.291101\pi\)
\(234\) 0 0
\(235\) 1.25544 0.0818957
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 22.3505 1.44574 0.722868 0.690986i \(-0.242824\pi\)
0.722868 + 0.690986i \(0.242824\pi\)
\(240\) 0 0
\(241\) −24.9783 −1.60899 −0.804495 0.593959i \(-0.797564\pi\)
−0.804495 + 0.593959i \(0.797564\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 4.37228 0.279335
\(246\) 0 0
\(247\) 6.74456 0.429146
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −17.4891 −1.10390 −0.551952 0.833876i \(-0.686117\pi\)
−0.551952 + 0.833876i \(0.686117\pi\)
\(252\) 0 0
\(253\) 2.11684 0.133085
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −10.0000 −0.623783 −0.311891 0.950118i \(-0.600963\pi\)
−0.311891 + 0.950118i \(0.600963\pi\)
\(258\) 0 0
\(259\) 18.1168 1.12573
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 17.4891 1.07843 0.539213 0.842170i \(-0.318722\pi\)
0.539213 + 0.842170i \(0.318722\pi\)
\(264\) 0 0
\(265\) 9.37228 0.575735
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −12.7446 −0.777050 −0.388525 0.921438i \(-0.627015\pi\)
−0.388525 + 0.921438i \(0.627015\pi\)
\(270\) 0 0
\(271\) 17.2554 1.04819 0.524097 0.851659i \(-0.324403\pi\)
0.524097 + 0.851659i \(0.324403\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3.37228 0.203356
\(276\) 0 0
\(277\) −24.9783 −1.50080 −0.750399 0.660985i \(-0.770139\pi\)
−0.750399 + 0.660985i \(0.770139\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −12.9783 −0.774218 −0.387109 0.922034i \(-0.626526\pi\)
−0.387109 + 0.922034i \(0.626526\pi\)
\(282\) 0 0
\(283\) 20.0000 1.18888 0.594438 0.804141i \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 4.62772 0.273166
\(288\) 0 0
\(289\) −15.1168 −0.889226
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 15.2554 0.891232 0.445616 0.895224i \(-0.352985\pi\)
0.445616 + 0.895224i \(0.352985\pi\)
\(294\) 0 0
\(295\) 8.00000 0.465778
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −0.627719 −0.0363019
\(300\) 0 0
\(301\) −13.4891 −0.777500
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 8.11684 0.464769
\(306\) 0 0
\(307\) 15.3723 0.877342 0.438671 0.898648i \(-0.355449\pi\)
0.438671 + 0.898648i \(0.355449\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 18.2337 1.02411 0.512053 0.858954i \(-0.328885\pi\)
0.512053 + 0.858954i \(0.328885\pi\)
\(318\) 0 0
\(319\) 6.74456 0.377623
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −9.25544 −0.514986
\(324\) 0 0
\(325\) −1.00000 −0.0554700
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 4.23369 0.233411
\(330\) 0 0
\(331\) −21.4891 −1.18115 −0.590575 0.806983i \(-0.701099\pi\)
−0.590575 + 0.806983i \(0.701099\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) −18.2337 −0.993252 −0.496626 0.867965i \(-0.665428\pi\)
−0.496626 + 0.867965i \(0.665428\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 22.7446 1.23169
\(342\) 0 0
\(343\) −8.86141 −0.478471
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −20.8614 −1.11990 −0.559949 0.828527i \(-0.689179\pi\)
−0.559949 + 0.828527i \(0.689179\pi\)
\(348\) 0 0
\(349\) −2.23369 −0.119567 −0.0597833 0.998211i \(-0.519041\pi\)
−0.0597833 + 0.998211i \(0.519041\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −14.0000 −0.745145 −0.372572 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) 0 0
\(355\) 11.3723 0.603578
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −8.00000 −0.422224 −0.211112 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) 0 0
\(361\) 26.4891 1.39416
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −15.4891 −0.810738
\(366\) 0 0
\(367\) −9.48913 −0.495328 −0.247664 0.968846i \(-0.579663\pi\)
−0.247664 + 0.968846i \(0.579663\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 31.6060 1.64090
\(372\) 0 0
\(373\) −15.2554 −0.789897 −0.394948 0.918703i \(-0.629237\pi\)
−0.394948 + 0.918703i \(0.629237\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −2.00000 −0.103005
\(378\) 0 0
\(379\) 14.7446 0.757377 0.378689 0.925524i \(-0.376375\pi\)
0.378689 + 0.925524i \(0.376375\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −9.25544 −0.472931 −0.236465 0.971640i \(-0.575989\pi\)
−0.236465 + 0.971640i \(0.575989\pi\)
\(384\) 0 0
\(385\) 11.3723 0.579585
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −26.2337 −1.33010 −0.665050 0.746798i \(-0.731590\pi\)
−0.665050 + 0.746798i \(0.731590\pi\)
\(390\) 0 0
\(391\) 0.861407 0.0435632
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −16.8614 −0.848389
\(396\) 0 0
\(397\) 24.3505 1.22212 0.611059 0.791585i \(-0.290744\pi\)
0.611059 + 0.791585i \(0.290744\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −10.0000 −0.499376 −0.249688 0.968326i \(-0.580328\pi\)
−0.249688 + 0.968326i \(0.580328\pi\)
\(402\) 0 0
\(403\) −6.74456 −0.335971
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 18.1168 0.898019
\(408\) 0 0
\(409\) 12.9783 0.641733 0.320867 0.947124i \(-0.396026\pi\)
0.320867 + 0.947124i \(0.396026\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 26.9783 1.32751
\(414\) 0 0
\(415\) 12.0000 0.589057
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 32.2337 1.57472 0.787359 0.616494i \(-0.211448\pi\)
0.787359 + 0.616494i \(0.211448\pi\)
\(420\) 0 0
\(421\) 27.7228 1.35113 0.675564 0.737302i \(-0.263900\pi\)
0.675564 + 0.737302i \(0.263900\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1.37228 0.0665654
\(426\) 0 0
\(427\) 27.3723 1.32464
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −18.9783 −0.914150 −0.457075 0.889428i \(-0.651103\pi\)
−0.457075 + 0.889428i \(0.651103\pi\)
\(432\) 0 0
\(433\) 6.23369 0.299572 0.149786 0.988718i \(-0.452142\pi\)
0.149786 + 0.988718i \(0.452142\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −4.23369 −0.202525
\(438\) 0 0
\(439\) 0.394031 0.0188061 0.00940303 0.999956i \(-0.497007\pi\)
0.00940303 + 0.999956i \(0.497007\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −18.3505 −0.871860 −0.435930 0.899981i \(-0.643580\pi\)
−0.435930 + 0.899981i \(0.643580\pi\)
\(444\) 0 0
\(445\) −13.3723 −0.633907
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1.37228 0.0647620 0.0323810 0.999476i \(-0.489691\pi\)
0.0323810 + 0.999476i \(0.489691\pi\)
\(450\) 0 0
\(451\) 4.62772 0.217911
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −3.37228 −0.158095
\(456\) 0 0
\(457\) 40.1168 1.87659 0.938293 0.345840i \(-0.112406\pi\)
0.938293 + 0.345840i \(0.112406\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 20.3505 0.947819 0.473909 0.880574i \(-0.342842\pi\)
0.473909 + 0.880574i \(0.342842\pi\)
\(462\) 0 0
\(463\) −31.6060 −1.46885 −0.734427 0.678688i \(-0.762549\pi\)
−0.734427 + 0.678688i \(0.762549\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −10.3505 −0.478965 −0.239483 0.970901i \(-0.576978\pi\)
−0.239483 + 0.970901i \(0.576978\pi\)
\(468\) 0 0
\(469\) −13.4891 −0.622870
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −13.4891 −0.620231
\(474\) 0 0
\(475\) −6.74456 −0.309462
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −13.8832 −0.634338 −0.317169 0.948369i \(-0.602732\pi\)
−0.317169 + 0.948369i \(0.602732\pi\)
\(480\) 0 0
\(481\) −5.37228 −0.244955
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 2.62772 0.119319
\(486\) 0 0
\(487\) 2.11684 0.0959234 0.0479617 0.998849i \(-0.484727\pi\)
0.0479617 + 0.998849i \(0.484727\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 0 0
\(493\) 2.74456 0.123609
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 38.3505 1.72026
\(498\) 0 0
\(499\) −14.7446 −0.660057 −0.330029 0.943971i \(-0.607058\pi\)
−0.330029 + 0.943971i \(0.607058\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 22.9783 1.02455 0.512275 0.858822i \(-0.328803\pi\)
0.512275 + 0.858822i \(0.328803\pi\)
\(504\) 0 0
\(505\) −4.74456 −0.211130
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −34.8614 −1.54520 −0.772602 0.634890i \(-0.781045\pi\)
−0.772602 + 0.634890i \(0.781045\pi\)
\(510\) 0 0
\(511\) −52.2337 −2.31068
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −4.00000 −0.176261
\(516\) 0 0
\(517\) 4.23369 0.186197
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −23.4891 −1.02908 −0.514539 0.857467i \(-0.672037\pi\)
−0.514539 + 0.857467i \(0.672037\pi\)
\(522\) 0 0
\(523\) 28.0000 1.22435 0.612177 0.790721i \(-0.290294\pi\)
0.612177 + 0.790721i \(0.290294\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 9.25544 0.403173
\(528\) 0 0
\(529\) −22.6060 −0.982868
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −1.37228 −0.0594401
\(534\) 0 0
\(535\) 8.62772 0.373009
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 14.7446 0.635093
\(540\) 0 0
\(541\) 23.4891 1.00988 0.504938 0.863156i \(-0.331515\pi\)
0.504938 + 0.863156i \(0.331515\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 8.74456 0.374576
\(546\) 0 0
\(547\) 6.97825 0.298368 0.149184 0.988809i \(-0.452335\pi\)
0.149184 + 0.988809i \(0.452335\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −13.4891 −0.574656
\(552\) 0 0
\(553\) −56.8614 −2.41799
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −12.9783 −0.549906 −0.274953 0.961458i \(-0.588662\pi\)
−0.274953 + 0.961458i \(0.588662\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −1.88316 −0.0793656 −0.0396828 0.999212i \(-0.512635\pi\)
−0.0396828 + 0.999212i \(0.512635\pi\)
\(564\) 0 0
\(565\) 6.00000 0.252422
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 23.7228 0.994512 0.497256 0.867604i \(-0.334341\pi\)
0.497256 + 0.867604i \(0.334341\pi\)
\(570\) 0 0
\(571\) −34.3505 −1.43753 −0.718763 0.695256i \(-0.755291\pi\)
−0.718763 + 0.695256i \(0.755291\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0.627719 0.0261777
\(576\) 0 0
\(577\) 40.1168 1.67009 0.835043 0.550185i \(-0.185443\pi\)
0.835043 + 0.550185i \(0.185443\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 40.4674 1.67887
\(582\) 0 0
\(583\) 31.6060 1.30899
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 2.74456 0.113280 0.0566401 0.998395i \(-0.481961\pi\)
0.0566401 + 0.998395i \(0.481961\pi\)
\(588\) 0 0
\(589\) −45.4891 −1.87435
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 11.2554 0.462205 0.231103 0.972929i \(-0.425767\pi\)
0.231103 + 0.972929i \(0.425767\pi\)
\(594\) 0 0
\(595\) 4.62772 0.189718
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −5.48913 −0.224280 −0.112140 0.993692i \(-0.535770\pi\)
−0.112140 + 0.993692i \(0.535770\pi\)
\(600\) 0 0
\(601\) −37.6060 −1.53398 −0.766990 0.641659i \(-0.778246\pi\)
−0.766990 + 0.641659i \(0.778246\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0.372281 0.0151354
\(606\) 0 0
\(607\) 28.0000 1.13648 0.568242 0.822861i \(-0.307624\pi\)
0.568242 + 0.822861i \(0.307624\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −1.25544 −0.0507896
\(612\) 0 0
\(613\) −14.8614 −0.600247 −0.300123 0.953900i \(-0.597028\pi\)
−0.300123 + 0.953900i \(0.597028\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −46.4674 −1.87071 −0.935353 0.353716i \(-0.884918\pi\)
−0.935353 + 0.353716i \(0.884918\pi\)
\(618\) 0 0
\(619\) 41.7228 1.67698 0.838491 0.544916i \(-0.183438\pi\)
0.838491 + 0.544916i \(0.183438\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −45.0951 −1.80670
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 7.37228 0.293952
\(630\) 0 0
\(631\) −18.5109 −0.736906 −0.368453 0.929646i \(-0.620113\pi\)
−0.368453 + 0.929646i \(0.620113\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −16.2337 −0.644214
\(636\) 0 0
\(637\) −4.37228 −0.173236
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −22.2337 −0.878178 −0.439089 0.898444i \(-0.644699\pi\)
−0.439089 + 0.898444i \(0.644699\pi\)
\(642\) 0 0
\(643\) −6.11684 −0.241225 −0.120612 0.992700i \(-0.538486\pi\)
−0.120612 + 0.992700i \(0.538486\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 4.86141 0.191122 0.0955608 0.995424i \(-0.469536\pi\)
0.0955608 + 0.995424i \(0.469536\pi\)
\(648\) 0 0
\(649\) 26.9783 1.05899
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 30.0000 1.17399 0.586995 0.809590i \(-0.300311\pi\)
0.586995 + 0.809590i \(0.300311\pi\)
\(654\) 0 0
\(655\) 18.7446 0.732411
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 14.5109 0.565263 0.282632 0.959229i \(-0.408793\pi\)
0.282632 + 0.959229i \(0.408793\pi\)
\(660\) 0 0
\(661\) −34.2337 −1.33154 −0.665768 0.746159i \(-0.731896\pi\)
−0.665768 + 0.746159i \(0.731896\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −22.7446 −0.881996
\(666\) 0 0
\(667\) 1.25544 0.0486107
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 27.3723 1.05670
\(672\) 0 0
\(673\) −8.51087 −0.328070 −0.164035 0.986455i \(-0.552451\pi\)
−0.164035 + 0.986455i \(0.552451\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 17.3723 0.667671 0.333836 0.942631i \(-0.391657\pi\)
0.333836 + 0.942631i \(0.391657\pi\)
\(678\) 0 0
\(679\) 8.86141 0.340070
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −1.02175 −0.0390962 −0.0195481 0.999809i \(-0.506223\pi\)
−0.0195481 + 0.999809i \(0.506223\pi\)
\(684\) 0 0
\(685\) −11.4891 −0.438977
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −9.37228 −0.357055
\(690\) 0 0
\(691\) −33.7228 −1.28288 −0.641438 0.767175i \(-0.721662\pi\)
−0.641438 + 0.767175i \(0.721662\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 7.37228 0.279647
\(696\) 0 0
\(697\) 1.88316 0.0713296
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 16.7446 0.632433 0.316217 0.948687i \(-0.397587\pi\)
0.316217 + 0.948687i \(0.397587\pi\)
\(702\) 0 0
\(703\) −36.2337 −1.36658
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −16.0000 −0.601742
\(708\) 0 0
\(709\) −39.2554 −1.47427 −0.737134 0.675746i \(-0.763822\pi\)
−0.737134 + 0.675746i \(0.763822\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 4.23369 0.158553
\(714\) 0 0
\(715\) −3.37228 −0.126116
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −50.9783 −1.90117 −0.950584 0.310468i \(-0.899514\pi\)
−0.950584 + 0.310468i \(0.899514\pi\)
\(720\) 0 0
\(721\) −13.4891 −0.502361
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 2.00000 0.0742781
\(726\) 0 0
\(727\) −10.7446 −0.398494 −0.199247 0.979949i \(-0.563850\pi\)
−0.199247 + 0.979949i \(0.563850\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −5.48913 −0.203023
\(732\) 0 0
\(733\) −43.0951 −1.59175 −0.795877 0.605459i \(-0.792990\pi\)
−0.795877 + 0.605459i \(0.792990\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −13.4891 −0.496878
\(738\) 0 0
\(739\) 28.2337 1.03859 0.519296 0.854594i \(-0.326194\pi\)
0.519296 + 0.854594i \(0.326194\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 12.2337 0.448810 0.224405 0.974496i \(-0.427956\pi\)
0.224405 + 0.974496i \(0.427956\pi\)
\(744\) 0 0
\(745\) −13.3723 −0.489922
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 29.0951 1.06311
\(750\) 0 0
\(751\) 8.86141 0.323357 0.161679 0.986843i \(-0.448309\pi\)
0.161679 + 0.986843i \(0.448309\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −39.2554 −1.42676 −0.713382 0.700776i \(-0.752837\pi\)
−0.713382 + 0.700776i \(0.752837\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0.510875 0.0185192 0.00925960 0.999957i \(-0.497053\pi\)
0.00925960 + 0.999957i \(0.497053\pi\)
\(762\) 0 0
\(763\) 29.4891 1.06758
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −8.00000 −0.288863
\(768\) 0 0
\(769\) 6.23369 0.224793 0.112396 0.993663i \(-0.464147\pi\)
0.112396 + 0.993663i \(0.464147\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −16.7446 −0.602260 −0.301130 0.953583i \(-0.597364\pi\)
−0.301130 + 0.953583i \(0.597364\pi\)
\(774\) 0 0
\(775\) 6.74456 0.242272
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −9.25544 −0.331610
\(780\) 0 0
\(781\) 38.3505 1.37229
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 2.00000 0.0713831
\(786\) 0 0
\(787\) −6.97825 −0.248748 −0.124374 0.992235i \(-0.539692\pi\)
−0.124374 + 0.992235i \(0.539692\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 20.2337 0.719427
\(792\) 0 0
\(793\) −8.11684 −0.288238
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 10.6277 0.376453 0.188227 0.982126i \(-0.439726\pi\)
0.188227 + 0.982126i \(0.439726\pi\)
\(798\) 0 0
\(799\) 1.72281 0.0609488
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −52.2337 −1.84329
\(804\) 0 0
\(805\) 2.11684 0.0746089
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −34.4674 −1.21181 −0.605904 0.795538i \(-0.707189\pi\)
−0.605904 + 0.795538i \(0.707189\pi\)
\(810\) 0 0
\(811\) 29.4891 1.03550 0.517752 0.855531i \(-0.326769\pi\)
0.517752 + 0.855531i \(0.326769\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −0.627719 −0.0219880
\(816\) 0 0
\(817\) 26.9783 0.943850
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −39.0951 −1.36443 −0.682214 0.731152i \(-0.738983\pi\)
−0.682214 + 0.731152i \(0.738983\pi\)
\(822\) 0 0
\(823\) −44.0000 −1.53374 −0.766872 0.641800i \(-0.778188\pi\)
−0.766872 + 0.641800i \(0.778188\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 32.2337 1.12088 0.560438 0.828197i \(-0.310633\pi\)
0.560438 + 0.828197i \(0.310633\pi\)
\(828\) 0 0
\(829\) −20.9783 −0.728605 −0.364302 0.931281i \(-0.618693\pi\)
−0.364302 + 0.931281i \(0.618693\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 6.00000 0.207888
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 11.3723 0.392615 0.196307 0.980542i \(-0.437105\pi\)
0.196307 + 0.980542i \(0.437105\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1.00000 0.0344010
\(846\) 0 0
\(847\) 1.25544 0.0431373
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 3.37228 0.115600
\(852\) 0 0
\(853\) −3.88316 −0.132957 −0.0664784 0.997788i \(-0.521176\pi\)
−0.0664784 + 0.997788i \(0.521176\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 21.6060 0.738046 0.369023 0.929420i \(-0.379692\pi\)
0.369023 + 0.929420i \(0.379692\pi\)
\(858\) 0 0
\(859\) −30.1168 −1.02757 −0.513787 0.857918i \(-0.671758\pi\)
−0.513787 + 0.857918i \(0.671758\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −46.7446 −1.59120 −0.795602 0.605820i \(-0.792845\pi\)
−0.795602 + 0.605820i \(0.792845\pi\)
\(864\) 0 0
\(865\) −7.48913 −0.254638
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −56.8614 −1.92889
\(870\) 0 0
\(871\) 4.00000 0.135535
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 3.37228 0.114004
\(876\) 0 0
\(877\) 12.9783 0.438244 0.219122 0.975697i \(-0.429681\pi\)
0.219122 + 0.975697i \(0.429681\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 0 0
\(883\) −6.97825 −0.234837 −0.117418 0.993083i \(-0.537462\pi\)
−0.117418 + 0.993083i \(0.537462\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 50.3505 1.69061 0.845303 0.534288i \(-0.179420\pi\)
0.845303 + 0.534288i \(0.179420\pi\)
\(888\) 0 0
\(889\) −54.7446 −1.83607
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −8.46738 −0.283350
\(894\) 0 0
\(895\) −24.2337 −0.810043
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 13.4891 0.449888
\(900\) 0 0
\(901\) 12.8614 0.428476
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −5.37228 −0.178581
\(906\) 0 0
\(907\) 37.2554 1.23705 0.618523 0.785767i \(-0.287731\pi\)
0.618523 + 0.785767i \(0.287731\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 24.0000 0.795155 0.397578 0.917568i \(-0.369851\pi\)
0.397578 + 0.917568i \(0.369851\pi\)
\(912\) 0 0
\(913\) 40.4674 1.33927
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 63.2119 2.08744
\(918\) 0 0
\(919\) −48.8614 −1.61179 −0.805895 0.592059i \(-0.798315\pi\)
−0.805895 + 0.592059i \(0.798315\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −11.3723 −0.374323
\(924\) 0 0
\(925\) 5.37228 0.176640
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −10.3940 −0.341017 −0.170509 0.985356i \(-0.554541\pi\)
−0.170509 + 0.985356i \(0.554541\pi\)
\(930\) 0 0
\(931\) −29.4891 −0.966467
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 4.62772 0.151343
\(936\) 0 0
\(937\) −3.48913 −0.113985 −0.0569924 0.998375i \(-0.518151\pi\)
−0.0569924 + 0.998375i \(0.518151\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −37.3723 −1.21830 −0.609151 0.793054i \(-0.708490\pi\)
−0.609151 + 0.793054i \(0.708490\pi\)
\(942\) 0 0
\(943\) 0.861407 0.0280513
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 46.9783 1.52659 0.763294 0.646051i \(-0.223581\pi\)
0.763294 + 0.646051i \(0.223581\pi\)
\(948\) 0 0
\(949\) 15.4891 0.502798
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 60.3505 1.95495 0.977473 0.211062i \(-0.0676921\pi\)
0.977473 + 0.211062i \(0.0676921\pi\)
\(954\) 0 0
\(955\) 8.00000 0.258874
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −38.7446 −1.25113
\(960\) 0 0
\(961\) 14.4891 0.467391
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 9.37228 0.301704
\(966\) 0 0
\(967\) 34.9783 1.12482 0.562412 0.826857i \(-0.309873\pi\)
0.562412 + 0.826857i \(0.309873\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −6.51087 −0.208944 −0.104472 0.994528i \(-0.533315\pi\)
−0.104472 + 0.994528i \(0.533315\pi\)
\(972\) 0 0
\(973\) 24.8614 0.797020
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −60.7446 −1.94339 −0.971695 0.236237i \(-0.924086\pi\)
−0.971695 + 0.236237i \(0.924086\pi\)
\(978\) 0 0
\(979\) −45.0951 −1.44125
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −5.48913 −0.175076 −0.0875380 0.996161i \(-0.527900\pi\)
−0.0875380 + 0.996161i \(0.527900\pi\)
\(984\) 0 0
\(985\) 10.2337 0.326072
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −2.51087 −0.0798412
\(990\) 0 0
\(991\) −15.6060 −0.495740 −0.247870 0.968793i \(-0.579731\pi\)
−0.247870 + 0.968793i \(0.579731\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 21.4891 0.681251
\(996\) 0 0
\(997\) −23.7228 −0.751309 −0.375655 0.926760i \(-0.622582\pi\)
−0.375655 + 0.926760i \(0.622582\pi\)
\(998\) 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 4680.2.a.bf.1.2 2
3.2 odd 2 1560.2.a.n.1.2 2
4.3 odd 2 9360.2.a.co.1.1 2
12.11 even 2 3120.2.a.bd.1.1 2
15.14 odd 2 7800.2.a.bb.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1560.2.a.n.1.2 2 3.2 odd 2
3120.2.a.bd.1.1 2 12.11 even 2
4680.2.a.bf.1.2 2 1.1 even 1 trivial
7800.2.a.bb.1.1 2 15.14 odd 2
9360.2.a.co.1.1 2 4.3 odd 2