Properties

Label 3920.2.a.bp.1.2
Level $3920$
Weight $2$
Character 3920.1
Self dual yes
Analytic conductor $31.301$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3920.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.3013575923\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 3920.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.414214 q^{3} -1.00000 q^{5} -2.82843 q^{9} +O(q^{10})\) \(q+0.414214 q^{3} -1.00000 q^{5} -2.82843 q^{9} -0.828427 q^{11} -2.00000 q^{13} -0.414214 q^{15} +7.65685 q^{17} -5.65685 q^{19} +5.58579 q^{23} +1.00000 q^{25} -2.41421 q^{27} -7.82843 q^{29} +0.828427 q^{31} -0.343146 q^{33} +5.65685 q^{37} -0.828427 q^{39} -5.82843 q^{41} +6.89949 q^{43} +2.82843 q^{45} +11.6569 q^{47} +3.17157 q^{51} -5.65685 q^{53} +0.828427 q^{55} -2.34315 q^{57} -4.00000 q^{59} -6.65685 q^{61} +2.00000 q^{65} +12.8995 q^{67} +2.31371 q^{69} +12.0000 q^{71} -3.65685 q^{73} +0.414214 q^{75} +4.00000 q^{79} +7.48528 q^{81} -4.75736 q^{83} -7.65685 q^{85} -3.24264 q^{87} +5.34315 q^{89} +0.343146 q^{93} +5.65685 q^{95} -6.00000 q^{97} +2.34315 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 2q^{5} + O(q^{10}) \) \( 2q - 2q^{3} - 2q^{5} + 4q^{11} - 4q^{13} + 2q^{15} + 4q^{17} + 14q^{23} + 2q^{25} - 2q^{27} - 10q^{29} - 4q^{31} - 12q^{33} + 4q^{39} - 6q^{41} - 6q^{43} + 12q^{47} + 12q^{51} - 4q^{55} - 16q^{57} - 8q^{59} - 2q^{61} + 4q^{65} + 6q^{67} - 18q^{69} + 24q^{71} + 4q^{73} - 2q^{75} + 8q^{79} - 2q^{81} - 18q^{83} - 4q^{85} + 2q^{87} + 22q^{89} + 12q^{93} - 12q^{97} + 16q^{99} + O(q^{100}) \)

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
\(3\) 0.414214 0.239146 0.119573 0.992825i \(-0.461847\pi\)
0.119573 + 0.992825i \(0.461847\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −2.82843 −0.942809
\(10\) 0 0
\(11\) −0.828427 −0.249780 −0.124890 0.992171i \(-0.539858\pi\)
−0.124890 + 0.992171i \(0.539858\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) −0.414214 −0.106949
\(16\) 0 0
\(17\) 7.65685 1.85706 0.928530 0.371257i \(-0.121073\pi\)
0.928530 + 0.371257i \(0.121073\pi\)
\(18\) 0 0
\(19\) −5.65685 −1.29777 −0.648886 0.760886i \(-0.724765\pi\)
−0.648886 + 0.760886i \(0.724765\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.58579 1.16472 0.582358 0.812932i \(-0.302130\pi\)
0.582358 + 0.812932i \(0.302130\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −2.41421 −0.464616
\(28\) 0 0
\(29\) −7.82843 −1.45370 −0.726851 0.686795i \(-0.759017\pi\)
−0.726851 + 0.686795i \(0.759017\pi\)
\(30\) 0 0
\(31\) 0.828427 0.148790 0.0743950 0.997229i \(-0.476297\pi\)
0.0743950 + 0.997229i \(0.476297\pi\)
\(32\) 0 0
\(33\) −0.343146 −0.0597340
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.65685 0.929981 0.464991 0.885316i \(-0.346058\pi\)
0.464991 + 0.885316i \(0.346058\pi\)
\(38\) 0 0
\(39\) −0.828427 −0.132655
\(40\) 0 0
\(41\) −5.82843 −0.910247 −0.455124 0.890428i \(-0.650405\pi\)
−0.455124 + 0.890428i \(0.650405\pi\)
\(42\) 0 0
\(43\) 6.89949 1.05216 0.526082 0.850434i \(-0.323661\pi\)
0.526082 + 0.850434i \(0.323661\pi\)
\(44\) 0 0
\(45\) 2.82843 0.421637
\(46\) 0 0
\(47\) 11.6569 1.70033 0.850163 0.526519i \(-0.176503\pi\)
0.850163 + 0.526519i \(0.176503\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 3.17157 0.444109
\(52\) 0 0
\(53\) −5.65685 −0.777029 −0.388514 0.921443i \(-0.627012\pi\)
−0.388514 + 0.921443i \(0.627012\pi\)
\(54\) 0 0
\(55\) 0.828427 0.111705
\(56\) 0 0
\(57\) −2.34315 −0.310357
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −6.65685 −0.852323 −0.426161 0.904647i \(-0.640134\pi\)
−0.426161 + 0.904647i \(0.640134\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) 12.8995 1.57592 0.787962 0.615724i \(-0.211136\pi\)
0.787962 + 0.615724i \(0.211136\pi\)
\(68\) 0 0
\(69\) 2.31371 0.278538
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) −3.65685 −0.428002 −0.214001 0.976833i \(-0.568650\pi\)
−0.214001 + 0.976833i \(0.568650\pi\)
\(74\) 0 0
\(75\) 0.414214 0.0478293
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 7.48528 0.831698
\(82\) 0 0
\(83\) −4.75736 −0.522188 −0.261094 0.965313i \(-0.584083\pi\)
−0.261094 + 0.965313i \(0.584083\pi\)
\(84\) 0 0
\(85\) −7.65685 −0.830502
\(86\) 0 0
\(87\) −3.24264 −0.347648
\(88\) 0 0
\(89\) 5.34315 0.566372 0.283186 0.959065i \(-0.408609\pi\)
0.283186 + 0.959065i \(0.408609\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0.343146 0.0355826
\(94\) 0 0
\(95\) 5.65685 0.580381
\(96\) 0 0
\(97\) −6.00000 −0.609208 −0.304604 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 0 0
\(99\) 2.34315 0.235495
\(100\) 0 0
\(101\) 11.4853 1.14283 0.571414 0.820662i \(-0.306395\pi\)
0.571414 + 0.820662i \(0.306395\pi\)
\(102\) 0 0
\(103\) 7.58579 0.747450 0.373725 0.927540i \(-0.378080\pi\)
0.373725 + 0.927540i \(0.378080\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 5.58579 0.539998 0.269999 0.962861i \(-0.412977\pi\)
0.269999 + 0.962861i \(0.412977\pi\)
\(108\) 0 0
\(109\) 18.3137 1.75414 0.877068 0.480367i \(-0.159497\pi\)
0.877068 + 0.480367i \(0.159497\pi\)
\(110\) 0 0
\(111\) 2.34315 0.222402
\(112\) 0 0
\(113\) 11.3137 1.06430 0.532152 0.846649i \(-0.321383\pi\)
0.532152 + 0.846649i \(0.321383\pi\)
\(114\) 0 0
\(115\) −5.58579 −0.520877
\(116\) 0 0
\(117\) 5.65685 0.522976
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −10.3137 −0.937610
\(122\) 0 0
\(123\) −2.41421 −0.217682
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 4.34315 0.385392 0.192696 0.981259i \(-0.438277\pi\)
0.192696 + 0.981259i \(0.438277\pi\)
\(128\) 0 0
\(129\) 2.85786 0.251621
\(130\) 0 0
\(131\) 13.6569 1.19320 0.596602 0.802537i \(-0.296517\pi\)
0.596602 + 0.802537i \(0.296517\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 2.41421 0.207782
\(136\) 0 0
\(137\) −4.00000 −0.341743 −0.170872 0.985293i \(-0.554658\pi\)
−0.170872 + 0.985293i \(0.554658\pi\)
\(138\) 0 0
\(139\) 2.48528 0.210799 0.105399 0.994430i \(-0.466388\pi\)
0.105399 + 0.994430i \(0.466388\pi\)
\(140\) 0 0
\(141\) 4.82843 0.406627
\(142\) 0 0
\(143\) 1.65685 0.138553
\(144\) 0 0
\(145\) 7.82843 0.650115
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 4.65685 0.381504 0.190752 0.981638i \(-0.438907\pi\)
0.190752 + 0.981638i \(0.438907\pi\)
\(150\) 0 0
\(151\) 11.1716 0.909130 0.454565 0.890714i \(-0.349795\pi\)
0.454565 + 0.890714i \(0.349795\pi\)
\(152\) 0 0
\(153\) −21.6569 −1.75085
\(154\) 0 0
\(155\) −0.828427 −0.0665409
\(156\) 0 0
\(157\) −1.31371 −0.104845 −0.0524227 0.998625i \(-0.516694\pi\)
−0.0524227 + 0.998625i \(0.516694\pi\)
\(158\) 0 0
\(159\) −2.34315 −0.185824
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 15.6569 1.22634 0.613170 0.789951i \(-0.289894\pi\)
0.613170 + 0.789951i \(0.289894\pi\)
\(164\) 0 0
\(165\) 0.343146 0.0267139
\(166\) 0 0
\(167\) 2.07107 0.160264 0.0801320 0.996784i \(-0.474466\pi\)
0.0801320 + 0.996784i \(0.474466\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 16.0000 1.22355
\(172\) 0 0
\(173\) −10.3431 −0.786375 −0.393187 0.919458i \(-0.628628\pi\)
−0.393187 + 0.919458i \(0.628628\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −1.65685 −0.124537
\(178\) 0 0
\(179\) 6.48528 0.484733 0.242366 0.970185i \(-0.422076\pi\)
0.242366 + 0.970185i \(0.422076\pi\)
\(180\) 0 0
\(181\) 4.17157 0.310071 0.155035 0.987909i \(-0.450451\pi\)
0.155035 + 0.987909i \(0.450451\pi\)
\(182\) 0 0
\(183\) −2.75736 −0.203830
\(184\) 0 0
\(185\) −5.65685 −0.415900
\(186\) 0 0
\(187\) −6.34315 −0.463857
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 5.51472 0.399031 0.199516 0.979895i \(-0.436063\pi\)
0.199516 + 0.979895i \(0.436063\pi\)
\(192\) 0 0
\(193\) −5.31371 −0.382489 −0.191245 0.981542i \(-0.561252\pi\)
−0.191245 + 0.981542i \(0.561252\pi\)
\(194\) 0 0
\(195\) 0.828427 0.0593249
\(196\) 0 0
\(197\) 0.343146 0.0244481 0.0122241 0.999925i \(-0.496109\pi\)
0.0122241 + 0.999925i \(0.496109\pi\)
\(198\) 0 0
\(199\) −23.3137 −1.65266 −0.826332 0.563183i \(-0.809577\pi\)
−0.826332 + 0.563183i \(0.809577\pi\)
\(200\) 0 0
\(201\) 5.34315 0.376876
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 5.82843 0.407075
\(206\) 0 0
\(207\) −15.7990 −1.09811
\(208\) 0 0
\(209\) 4.68629 0.324158
\(210\) 0 0
\(211\) −26.6274 −1.83311 −0.916553 0.399912i \(-0.869041\pi\)
−0.916553 + 0.399912i \(0.869041\pi\)
\(212\) 0 0
\(213\) 4.97056 0.340577
\(214\) 0 0
\(215\) −6.89949 −0.470542
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −1.51472 −0.102355
\(220\) 0 0
\(221\) −15.3137 −1.03011
\(222\) 0 0
\(223\) −14.9706 −1.00250 −0.501252 0.865302i \(-0.667127\pi\)
−0.501252 + 0.865302i \(0.667127\pi\)
\(224\) 0 0
\(225\) −2.82843 −0.188562
\(226\) 0 0
\(227\) 14.0000 0.929213 0.464606 0.885517i \(-0.346196\pi\)
0.464606 + 0.885517i \(0.346196\pi\)
\(228\) 0 0
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 11.6569 0.763666 0.381833 0.924231i \(-0.375293\pi\)
0.381833 + 0.924231i \(0.375293\pi\)
\(234\) 0 0
\(235\) −11.6569 −0.760409
\(236\) 0 0
\(237\) 1.65685 0.107624
\(238\) 0 0
\(239\) 30.4853 1.97193 0.985964 0.166955i \(-0.0533936\pi\)
0.985964 + 0.166955i \(0.0533936\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 0 0
\(243\) 10.3431 0.663513
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 11.3137 0.719874
\(248\) 0 0
\(249\) −1.97056 −0.124879
\(250\) 0 0
\(251\) −27.4558 −1.73300 −0.866499 0.499179i \(-0.833635\pi\)
−0.866499 + 0.499179i \(0.833635\pi\)
\(252\) 0 0
\(253\) −4.62742 −0.290923
\(254\) 0 0
\(255\) −3.17157 −0.198612
\(256\) 0 0
\(257\) −15.3137 −0.955243 −0.477621 0.878566i \(-0.658501\pi\)
−0.477621 + 0.878566i \(0.658501\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 22.1421 1.37056
\(262\) 0 0
\(263\) 15.7279 0.969825 0.484913 0.874563i \(-0.338851\pi\)
0.484913 + 0.874563i \(0.338851\pi\)
\(264\) 0 0
\(265\) 5.65685 0.347498
\(266\) 0 0
\(267\) 2.21320 0.135446
\(268\) 0 0
\(269\) 6.65685 0.405876 0.202938 0.979192i \(-0.434951\pi\)
0.202938 + 0.979192i \(0.434951\pi\)
\(270\) 0 0
\(271\) −3.31371 −0.201293 −0.100647 0.994922i \(-0.532091\pi\)
−0.100647 + 0.994922i \(0.532091\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −0.828427 −0.0499560
\(276\) 0 0
\(277\) 28.6274 1.72005 0.860027 0.510248i \(-0.170446\pi\)
0.860027 + 0.510248i \(0.170446\pi\)
\(278\) 0 0
\(279\) −2.34315 −0.140280
\(280\) 0 0
\(281\) 2.68629 0.160251 0.0801254 0.996785i \(-0.474468\pi\)
0.0801254 + 0.996785i \(0.474468\pi\)
\(282\) 0 0
\(283\) 18.0000 1.06999 0.534994 0.844856i \(-0.320314\pi\)
0.534994 + 0.844856i \(0.320314\pi\)
\(284\) 0 0
\(285\) 2.34315 0.138796
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 41.6274 2.44867
\(290\) 0 0
\(291\) −2.48528 −0.145690
\(292\) 0 0
\(293\) −16.9706 −0.991431 −0.495715 0.868485i \(-0.665094\pi\)
−0.495715 + 0.868485i \(0.665094\pi\)
\(294\) 0 0
\(295\) 4.00000 0.232889
\(296\) 0 0
\(297\) 2.00000 0.116052
\(298\) 0 0
\(299\) −11.1716 −0.646069
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 4.75736 0.273303
\(304\) 0 0
\(305\) 6.65685 0.381170
\(306\) 0 0
\(307\) 4.75736 0.271517 0.135758 0.990742i \(-0.456653\pi\)
0.135758 + 0.990742i \(0.456653\pi\)
\(308\) 0 0
\(309\) 3.14214 0.178750
\(310\) 0 0
\(311\) 21.6569 1.22805 0.614024 0.789288i \(-0.289550\pi\)
0.614024 + 0.789288i \(0.289550\pi\)
\(312\) 0 0
\(313\) −20.9706 −1.18533 −0.592663 0.805450i \(-0.701923\pi\)
−0.592663 + 0.805450i \(0.701923\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −22.0000 −1.23564 −0.617822 0.786318i \(-0.711985\pi\)
−0.617822 + 0.786318i \(0.711985\pi\)
\(318\) 0 0
\(319\) 6.48528 0.363106
\(320\) 0 0
\(321\) 2.31371 0.129139
\(322\) 0 0
\(323\) −43.3137 −2.41004
\(324\) 0 0
\(325\) −2.00000 −0.110940
\(326\) 0 0
\(327\) 7.58579 0.419495
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −26.4853 −1.45576 −0.727881 0.685703i \(-0.759495\pi\)
−0.727881 + 0.685703i \(0.759495\pi\)
\(332\) 0 0
\(333\) −16.0000 −0.876795
\(334\) 0 0
\(335\) −12.8995 −0.704775
\(336\) 0 0
\(337\) 24.9706 1.36023 0.680117 0.733104i \(-0.261929\pi\)
0.680117 + 0.733104i \(0.261929\pi\)
\(338\) 0 0
\(339\) 4.68629 0.254524
\(340\) 0 0
\(341\) −0.686292 −0.0371648
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −2.31371 −0.124566
\(346\) 0 0
\(347\) −11.3848 −0.611167 −0.305583 0.952165i \(-0.598851\pi\)
−0.305583 + 0.952165i \(0.598851\pi\)
\(348\) 0 0
\(349\) −9.82843 −0.526104 −0.263052 0.964782i \(-0.584729\pi\)
−0.263052 + 0.964782i \(0.584729\pi\)
\(350\) 0 0
\(351\) 4.82843 0.257722
\(352\) 0 0
\(353\) −33.6569 −1.79137 −0.895687 0.444685i \(-0.853315\pi\)
−0.895687 + 0.444685i \(0.853315\pi\)
\(354\) 0 0
\(355\) −12.0000 −0.636894
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 6.48528 0.342280 0.171140 0.985247i \(-0.445255\pi\)
0.171140 + 0.985247i \(0.445255\pi\)
\(360\) 0 0
\(361\) 13.0000 0.684211
\(362\) 0 0
\(363\) −4.27208 −0.224226
\(364\) 0 0
\(365\) 3.65685 0.191408
\(366\) 0 0
\(367\) 21.5858 1.12677 0.563384 0.826195i \(-0.309499\pi\)
0.563384 + 0.826195i \(0.309499\pi\)
\(368\) 0 0
\(369\) 16.4853 0.858189
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 12.0000 0.621336 0.310668 0.950518i \(-0.399447\pi\)
0.310668 + 0.950518i \(0.399447\pi\)
\(374\) 0 0
\(375\) −0.414214 −0.0213899
\(376\) 0 0
\(377\) 15.6569 0.806369
\(378\) 0 0
\(379\) −4.68629 −0.240719 −0.120359 0.992730i \(-0.538405\pi\)
−0.120359 + 0.992730i \(0.538405\pi\)
\(380\) 0 0
\(381\) 1.79899 0.0921650
\(382\) 0 0
\(383\) −0.899495 −0.0459620 −0.0229810 0.999736i \(-0.507316\pi\)
−0.0229810 + 0.999736i \(0.507316\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −19.5147 −0.991989
\(388\) 0 0
\(389\) 5.31371 0.269416 0.134708 0.990885i \(-0.456990\pi\)
0.134708 + 0.990885i \(0.456990\pi\)
\(390\) 0 0
\(391\) 42.7696 2.16295
\(392\) 0 0
\(393\) 5.65685 0.285351
\(394\) 0 0
\(395\) −4.00000 −0.201262
\(396\) 0 0
\(397\) 24.6274 1.23601 0.618007 0.786172i \(-0.287940\pi\)
0.618007 + 0.786172i \(0.287940\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −32.3137 −1.61367 −0.806835 0.590777i \(-0.798821\pi\)
−0.806835 + 0.590777i \(0.798821\pi\)
\(402\) 0 0
\(403\) −1.65685 −0.0825338
\(404\) 0 0
\(405\) −7.48528 −0.371947
\(406\) 0 0
\(407\) −4.68629 −0.232291
\(408\) 0 0
\(409\) 25.1421 1.24320 0.621599 0.783335i \(-0.286483\pi\)
0.621599 + 0.783335i \(0.286483\pi\)
\(410\) 0 0
\(411\) −1.65685 −0.0817266
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 4.75736 0.233530
\(416\) 0 0
\(417\) 1.02944 0.0504118
\(418\) 0 0
\(419\) −15.3137 −0.748124 −0.374062 0.927404i \(-0.622035\pi\)
−0.374062 + 0.927404i \(0.622035\pi\)
\(420\) 0 0
\(421\) 27.3431 1.33262 0.666312 0.745673i \(-0.267872\pi\)
0.666312 + 0.745673i \(0.267872\pi\)
\(422\) 0 0
\(423\) −32.9706 −1.60308
\(424\) 0 0
\(425\) 7.65685 0.371412
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0.686292 0.0331345
\(430\) 0 0
\(431\) 0.828427 0.0399039 0.0199520 0.999801i \(-0.493649\pi\)
0.0199520 + 0.999801i \(0.493649\pi\)
\(432\) 0 0
\(433\) 19.3137 0.928158 0.464079 0.885794i \(-0.346385\pi\)
0.464079 + 0.885794i \(0.346385\pi\)
\(434\) 0 0
\(435\) 3.24264 0.155473
\(436\) 0 0
\(437\) −31.5980 −1.51154
\(438\) 0 0
\(439\) −18.3431 −0.875471 −0.437735 0.899104i \(-0.644219\pi\)
−0.437735 + 0.899104i \(0.644219\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 15.5858 0.740503 0.370252 0.928932i \(-0.379271\pi\)
0.370252 + 0.928932i \(0.379271\pi\)
\(444\) 0 0
\(445\) −5.34315 −0.253289
\(446\) 0 0
\(447\) 1.92893 0.0912354
\(448\) 0 0
\(449\) −7.48528 −0.353252 −0.176626 0.984278i \(-0.556518\pi\)
−0.176626 + 0.984278i \(0.556518\pi\)
\(450\) 0 0
\(451\) 4.82843 0.227362
\(452\) 0 0
\(453\) 4.62742 0.217415
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 28.9706 1.35519 0.677593 0.735437i \(-0.263023\pi\)
0.677593 + 0.735437i \(0.263023\pi\)
\(458\) 0 0
\(459\) −18.4853 −0.862819
\(460\) 0 0
\(461\) −1.31371 −0.0611855 −0.0305928 0.999532i \(-0.509739\pi\)
−0.0305928 + 0.999532i \(0.509739\pi\)
\(462\) 0 0
\(463\) −14.8995 −0.692438 −0.346219 0.938154i \(-0.612535\pi\)
−0.346219 + 0.938154i \(0.612535\pi\)
\(464\) 0 0
\(465\) −0.343146 −0.0159130
\(466\) 0 0
\(467\) −37.8701 −1.75242 −0.876209 0.481932i \(-0.839935\pi\)
−0.876209 + 0.481932i \(0.839935\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −0.544156 −0.0250734
\(472\) 0 0
\(473\) −5.71573 −0.262809
\(474\) 0 0
\(475\) −5.65685 −0.259554
\(476\) 0 0
\(477\) 16.0000 0.732590
\(478\) 0 0
\(479\) −1.51472 −0.0692093 −0.0346046 0.999401i \(-0.511017\pi\)
−0.0346046 + 0.999401i \(0.511017\pi\)
\(480\) 0 0
\(481\) −11.3137 −0.515861
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 6.00000 0.272446
\(486\) 0 0
\(487\) 38.2843 1.73483 0.867413 0.497589i \(-0.165781\pi\)
0.867413 + 0.497589i \(0.165781\pi\)
\(488\) 0 0
\(489\) 6.48528 0.293275
\(490\) 0 0
\(491\) −1.51472 −0.0683583 −0.0341791 0.999416i \(-0.510882\pi\)
−0.0341791 + 0.999416i \(0.510882\pi\)
\(492\) 0 0
\(493\) −59.9411 −2.69961
\(494\) 0 0
\(495\) −2.34315 −0.105317
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −44.1421 −1.97607 −0.988037 0.154219i \(-0.950714\pi\)
−0.988037 + 0.154219i \(0.950714\pi\)
\(500\) 0 0
\(501\) 0.857864 0.0383266
\(502\) 0 0
\(503\) −3.92893 −0.175182 −0.0875912 0.996157i \(-0.527917\pi\)
−0.0875912 + 0.996157i \(0.527917\pi\)
\(504\) 0 0
\(505\) −11.4853 −0.511088
\(506\) 0 0
\(507\) −3.72792 −0.165563
\(508\) 0 0
\(509\) 33.4853 1.48421 0.742105 0.670284i \(-0.233828\pi\)
0.742105 + 0.670284i \(0.233828\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 13.6569 0.602965
\(514\) 0 0
\(515\) −7.58579 −0.334270
\(516\) 0 0
\(517\) −9.65685 −0.424708
\(518\) 0 0
\(519\) −4.28427 −0.188059
\(520\) 0 0
\(521\) −36.6274 −1.60468 −0.802338 0.596870i \(-0.796411\pi\)
−0.802338 + 0.596870i \(0.796411\pi\)
\(522\) 0 0
\(523\) 27.9411 1.22178 0.610890 0.791715i \(-0.290812\pi\)
0.610890 + 0.791715i \(0.290812\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6.34315 0.276312
\(528\) 0 0
\(529\) 8.20101 0.356566
\(530\) 0 0
\(531\) 11.3137 0.490973
\(532\) 0 0
\(533\) 11.6569 0.504914
\(534\) 0 0
\(535\) −5.58579 −0.241495
\(536\) 0 0
\(537\) 2.68629 0.115922
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −23.4853 −1.00971 −0.504856 0.863204i \(-0.668454\pi\)
−0.504856 + 0.863204i \(0.668454\pi\)
\(542\) 0 0
\(543\) 1.72792 0.0741522
\(544\) 0 0
\(545\) −18.3137 −0.784473
\(546\) 0 0
\(547\) 2.27208 0.0971470 0.0485735 0.998820i \(-0.484532\pi\)
0.0485735 + 0.998820i \(0.484532\pi\)
\(548\) 0 0
\(549\) 18.8284 0.803578
\(550\) 0 0
\(551\) 44.2843 1.88657
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −2.34315 −0.0994610
\(556\) 0 0
\(557\) −17.3137 −0.733605 −0.366803 0.930299i \(-0.619548\pi\)
−0.366803 + 0.930299i \(0.619548\pi\)
\(558\) 0 0
\(559\) −13.7990 −0.583635
\(560\) 0 0
\(561\) −2.62742 −0.110930
\(562\) 0 0
\(563\) −4.07107 −0.171575 −0.0857875 0.996313i \(-0.527341\pi\)
−0.0857875 + 0.996313i \(0.527341\pi\)
\(564\) 0 0
\(565\) −11.3137 −0.475971
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −36.6274 −1.53550 −0.767751 0.640749i \(-0.778624\pi\)
−0.767751 + 0.640749i \(0.778624\pi\)
\(570\) 0 0
\(571\) 20.9706 0.877591 0.438795 0.898587i \(-0.355405\pi\)
0.438795 + 0.898587i \(0.355405\pi\)
\(572\) 0 0
\(573\) 2.28427 0.0954268
\(574\) 0 0
\(575\) 5.58579 0.232943
\(576\) 0 0
\(577\) 22.2843 0.927706 0.463853 0.885912i \(-0.346467\pi\)
0.463853 + 0.885912i \(0.346467\pi\)
\(578\) 0 0
\(579\) −2.20101 −0.0914709
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 4.68629 0.194086
\(584\) 0 0
\(585\) −5.65685 −0.233882
\(586\) 0 0
\(587\) −22.6863 −0.936363 −0.468182 0.883632i \(-0.655091\pi\)
−0.468182 + 0.883632i \(0.655091\pi\)
\(588\) 0 0
\(589\) −4.68629 −0.193095
\(590\) 0 0
\(591\) 0.142136 0.00584668
\(592\) 0 0
\(593\) −29.9411 −1.22953 −0.614767 0.788709i \(-0.710750\pi\)
−0.614767 + 0.788709i \(0.710750\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −9.65685 −0.395229
\(598\) 0 0
\(599\) −42.6274 −1.74171 −0.870855 0.491541i \(-0.836434\pi\)
−0.870855 + 0.491541i \(0.836434\pi\)
\(600\) 0 0
\(601\) 34.0000 1.38689 0.693444 0.720510i \(-0.256092\pi\)
0.693444 + 0.720510i \(0.256092\pi\)
\(602\) 0 0
\(603\) −36.4853 −1.48580
\(604\) 0 0
\(605\) 10.3137 0.419312
\(606\) 0 0
\(607\) 27.2426 1.10574 0.552872 0.833266i \(-0.313532\pi\)
0.552872 + 0.833266i \(0.313532\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −23.3137 −0.943172
\(612\) 0 0
\(613\) 38.9706 1.57401 0.787003 0.616949i \(-0.211632\pi\)
0.787003 + 0.616949i \(0.211632\pi\)
\(614\) 0 0
\(615\) 2.41421 0.0973505
\(616\) 0 0
\(617\) −12.6863 −0.510731 −0.255365 0.966845i \(-0.582196\pi\)
−0.255365 + 0.966845i \(0.582196\pi\)
\(618\) 0 0
\(619\) 39.4558 1.58586 0.792932 0.609310i \(-0.208553\pi\)
0.792932 + 0.609310i \(0.208553\pi\)
\(620\) 0 0
\(621\) −13.4853 −0.541146
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 1.94113 0.0775211
\(628\) 0 0
\(629\) 43.3137 1.72703
\(630\) 0 0
\(631\) 1.51472 0.0603000 0.0301500 0.999545i \(-0.490402\pi\)
0.0301500 + 0.999545i \(0.490402\pi\)
\(632\) 0 0
\(633\) −11.0294 −0.438381
\(634\) 0 0
\(635\) −4.34315 −0.172352
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −33.9411 −1.34269
\(640\) 0 0
\(641\) 14.1127 0.557418 0.278709 0.960376i \(-0.410093\pi\)
0.278709 + 0.960376i \(0.410093\pi\)
\(642\) 0 0
\(643\) 26.0000 1.02534 0.512670 0.858586i \(-0.328656\pi\)
0.512670 + 0.858586i \(0.328656\pi\)
\(644\) 0 0
\(645\) −2.85786 −0.112528
\(646\) 0 0
\(647\) −20.7574 −0.816056 −0.408028 0.912969i \(-0.633783\pi\)
−0.408028 + 0.912969i \(0.633783\pi\)
\(648\) 0 0
\(649\) 3.31371 0.130074
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −15.6569 −0.612700 −0.306350 0.951919i \(-0.599108\pi\)
−0.306350 + 0.951919i \(0.599108\pi\)
\(654\) 0 0
\(655\) −13.6569 −0.533617
\(656\) 0 0
\(657\) 10.3431 0.403525
\(658\) 0 0
\(659\) −12.6863 −0.494188 −0.247094 0.968992i \(-0.579476\pi\)
−0.247094 + 0.968992i \(0.579476\pi\)
\(660\) 0 0
\(661\) 50.3137 1.95698 0.978488 0.206303i \(-0.0661431\pi\)
0.978488 + 0.206303i \(0.0661431\pi\)
\(662\) 0 0
\(663\) −6.34315 −0.246347
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −43.7279 −1.69315
\(668\) 0 0
\(669\) −6.20101 −0.239745
\(670\) 0 0
\(671\) 5.51472 0.212893
\(672\) 0 0
\(673\) −5.65685 −0.218056 −0.109028 0.994039i \(-0.534774\pi\)
−0.109028 + 0.994039i \(0.534774\pi\)
\(674\) 0 0
\(675\) −2.41421 −0.0929231
\(676\) 0 0
\(677\) −22.9706 −0.882830 −0.441415 0.897303i \(-0.645523\pi\)
−0.441415 + 0.897303i \(0.645523\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 5.79899 0.222218
\(682\) 0 0
\(683\) 1.92893 0.0738085 0.0369043 0.999319i \(-0.488250\pi\)
0.0369043 + 0.999319i \(0.488250\pi\)
\(684\) 0 0
\(685\) 4.00000 0.152832
\(686\) 0 0
\(687\) 5.79899 0.221245
\(688\) 0 0
\(689\) 11.3137 0.431018
\(690\) 0 0
\(691\) −42.7696 −1.62703 −0.813515 0.581544i \(-0.802449\pi\)
−0.813515 + 0.581544i \(0.802449\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −2.48528 −0.0942721
\(696\) 0 0
\(697\) −44.6274 −1.69038
\(698\) 0 0
\(699\) 4.82843 0.182628
\(700\) 0 0
\(701\) −11.0000 −0.415464 −0.207732 0.978186i \(-0.566608\pi\)
−0.207732 + 0.978186i \(0.566608\pi\)
\(702\) 0 0
\(703\) −32.0000 −1.20690
\(704\) 0 0
\(705\) −4.82843 −0.181849
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −35.4264 −1.33047 −0.665233 0.746636i \(-0.731668\pi\)
−0.665233 + 0.746636i \(0.731668\pi\)
\(710\) 0 0
\(711\) −11.3137 −0.424297
\(712\) 0 0
\(713\) 4.62742 0.173298
\(714\) 0 0
\(715\) −1.65685 −0.0619628
\(716\) 0 0
\(717\) 12.6274 0.471580
\(718\) 0 0
\(719\) −25.7990 −0.962140 −0.481070 0.876682i \(-0.659752\pi\)
−0.481070 + 0.876682i \(0.659752\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 4.14214 0.154048
\(724\) 0 0
\(725\) −7.82843 −0.290740
\(726\) 0 0
\(727\) 38.0711 1.41198 0.705989 0.708223i \(-0.250503\pi\)
0.705989 + 0.708223i \(0.250503\pi\)
\(728\) 0 0
\(729\) −18.1716 −0.673021
\(730\) 0 0
\(731\) 52.8284 1.95393
\(732\) 0 0
\(733\) −7.65685 −0.282812 −0.141406 0.989952i \(-0.545162\pi\)
−0.141406 + 0.989952i \(0.545162\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −10.6863 −0.393635
\(738\) 0 0
\(739\) −16.8284 −0.619044 −0.309522 0.950892i \(-0.600169\pi\)
−0.309522 + 0.950892i \(0.600169\pi\)
\(740\) 0 0
\(741\) 4.68629 0.172155
\(742\) 0 0
\(743\) 10.7574 0.394649 0.197325 0.980338i \(-0.436775\pi\)
0.197325 + 0.980338i \(0.436775\pi\)
\(744\) 0 0
\(745\) −4.65685 −0.170614
\(746\) 0 0
\(747\) 13.4558 0.492324
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 12.0000 0.437886 0.218943 0.975738i \(-0.429739\pi\)
0.218943 + 0.975738i \(0.429739\pi\)
\(752\) 0 0
\(753\) −11.3726 −0.414440
\(754\) 0 0
\(755\) −11.1716 −0.406575
\(756\) 0 0
\(757\) 6.00000 0.218074 0.109037 0.994038i \(-0.465223\pi\)
0.109037 + 0.994038i \(0.465223\pi\)
\(758\) 0 0
\(759\) −1.91674 −0.0695732
\(760\) 0 0
\(761\) 43.9411 1.59286 0.796432 0.604728i \(-0.206718\pi\)
0.796432 + 0.604728i \(0.206718\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 21.6569 0.783005
\(766\) 0 0
\(767\) 8.00000 0.288863
\(768\) 0 0
\(769\) −43.2548 −1.55981 −0.779905 0.625898i \(-0.784732\pi\)
−0.779905 + 0.625898i \(0.784732\pi\)
\(770\) 0 0
\(771\) −6.34315 −0.228443
\(772\) 0 0
\(773\) −24.3431 −0.875562 −0.437781 0.899082i \(-0.644235\pi\)
−0.437781 + 0.899082i \(0.644235\pi\)
\(774\) 0 0
\(775\) 0.828427 0.0297580
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 32.9706 1.18129
\(780\) 0 0
\(781\) −9.94113 −0.355721
\(782\) 0 0
\(783\) 18.8995 0.675413
\(784\) 0 0
\(785\) 1.31371 0.0468883
\(786\) 0 0
\(787\) −1.58579 −0.0565272 −0.0282636 0.999601i \(-0.508998\pi\)
−0.0282636 + 0.999601i \(0.508998\pi\)
\(788\) 0 0
\(789\) 6.51472 0.231930
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 13.3137 0.472784
\(794\) 0 0
\(795\) 2.34315 0.0831028
\(796\) 0 0
\(797\) 35.3137 1.25088 0.625438 0.780274i \(-0.284920\pi\)
0.625438 + 0.780274i \(0.284920\pi\)
\(798\) 0 0
\(799\) 89.2548 3.15761
\(800\) 0 0
\(801\) −15.1127 −0.533981
\(802\) 0 0
\(803\) 3.02944 0.106907
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 2.75736 0.0970636
\(808\) 0 0
\(809\) −35.9706 −1.26466 −0.632329 0.774700i \(-0.717901\pi\)
−0.632329 + 0.774700i \(0.717901\pi\)
\(810\) 0 0
\(811\) 52.1421 1.83096 0.915479 0.402366i \(-0.131812\pi\)
0.915479 + 0.402366i \(0.131812\pi\)
\(812\) 0 0
\(813\) −1.37258 −0.0481386
\(814\) 0 0
\(815\) −15.6569 −0.548436
\(816\) 0 0
\(817\) −39.0294 −1.36547
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −32.6274 −1.13870 −0.569352 0.822094i \(-0.692806\pi\)
−0.569352 + 0.822094i \(0.692806\pi\)
\(822\) 0 0
\(823\) −30.0122 −1.04616 −0.523080 0.852284i \(-0.675217\pi\)
−0.523080 + 0.852284i \(0.675217\pi\)
\(824\) 0 0
\(825\) −0.343146 −0.0119468
\(826\) 0 0
\(827\) −13.5269 −0.470377 −0.235188 0.971950i \(-0.575571\pi\)
−0.235188 + 0.971950i \(0.575571\pi\)
\(828\) 0 0
\(829\) −5.31371 −0.184553 −0.0922764 0.995733i \(-0.529414\pi\)
−0.0922764 + 0.995733i \(0.529414\pi\)
\(830\) 0 0
\(831\) 11.8579 0.411345
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −2.07107 −0.0716723
\(836\) 0 0
\(837\) −2.00000 −0.0691301
\(838\) 0 0
\(839\) 20.1421 0.695384 0.347692 0.937609i \(-0.386966\pi\)
0.347692 + 0.937609i \(0.386966\pi\)
\(840\) 0 0
\(841\) 32.2843 1.11325
\(842\) 0 0
\(843\) 1.11270 0.0383234
\(844\) 0 0
\(845\) 9.00000 0.309609
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 7.45584 0.255884
\(850\) 0 0
\(851\) 31.5980 1.08316
\(852\) 0 0
\(853\) −12.6863 −0.434370 −0.217185 0.976130i \(-0.569688\pi\)
−0.217185 + 0.976130i \(0.569688\pi\)
\(854\) 0 0
\(855\) −16.0000 −0.547188
\(856\) 0 0
\(857\) 18.9706 0.648022 0.324011 0.946053i \(-0.394969\pi\)
0.324011 + 0.946053i \(0.394969\pi\)
\(858\) 0 0
\(859\) −30.6274 −1.04499 −0.522497 0.852641i \(-0.674999\pi\)
−0.522497 + 0.852641i \(0.674999\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 56.0122 1.90668 0.953339 0.301903i \(-0.0976219\pi\)
0.953339 + 0.301903i \(0.0976219\pi\)
\(864\) 0 0
\(865\) 10.3431 0.351678
\(866\) 0 0
\(867\) 17.2426 0.585591
\(868\) 0 0
\(869\) −3.31371 −0.112410
\(870\) 0 0
\(871\) −25.7990 −0.874165
\(872\) 0 0
\(873\) 16.9706 0.574367
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −28.2843 −0.955092 −0.477546 0.878607i \(-0.658474\pi\)
−0.477546 + 0.878607i \(0.658474\pi\)
\(878\) 0 0
\(879\) −7.02944 −0.237097
\(880\) 0 0
\(881\) 26.4558 0.891320 0.445660 0.895202i \(-0.352969\pi\)
0.445660 + 0.895202i \(0.352969\pi\)
\(882\) 0 0
\(883\) −6.68629 −0.225012 −0.112506 0.993651i \(-0.535888\pi\)
−0.112506 + 0.993651i \(0.535888\pi\)
\(884\) 0 0
\(885\) 1.65685 0.0556945
\(886\) 0 0
\(887\) 17.5858 0.590473 0.295236 0.955424i \(-0.404602\pi\)
0.295236 + 0.955424i \(0.404602\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −6.20101 −0.207742
\(892\) 0 0
\(893\) −65.9411 −2.20664
\(894\) 0 0
\(895\) −6.48528 −0.216779
\(896\) 0 0
\(897\) −4.62742 −0.154505
\(898\) 0 0
\(899\) −6.48528 −0.216296
\(900\) 0 0
\(901\) −43.3137 −1.44299
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −4.17157 −0.138668
\(906\) 0 0
\(907\) −46.2132 −1.53448 −0.767242 0.641358i \(-0.778372\pi\)
−0.767242 + 0.641358i \(0.778372\pi\)
\(908\) 0 0
\(909\) −32.4853 −1.07747
\(910\) 0 0
\(911\) 18.4853 0.612445 0.306222 0.951960i \(-0.400935\pi\)
0.306222 + 0.951960i \(0.400935\pi\)
\(912\) 0 0
\(913\) 3.94113 0.130432
\(914\) 0 0
\(915\) 2.75736 0.0911555
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 33.2548 1.09698 0.548488 0.836159i \(-0.315204\pi\)
0.548488 + 0.836159i \(0.315204\pi\)
\(920\) 0 0
\(921\) 1.97056 0.0649323
\(922\) 0 0
\(923\) −24.0000 −0.789970
\(924\) 0 0
\(925\) 5.65685 0.185996
\(926\) 0 0
\(927\) −21.4558 −0.704702
\(928\) 0 0
\(929\) −9.20101 −0.301875 −0.150938 0.988543i \(-0.548229\pi\)
−0.150938 + 0.988543i \(0.548229\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 8.97056 0.293683
\(934\) 0 0
\(935\) 6.34315 0.207443
\(936\) 0 0
\(937\) −18.6274 −0.608531 −0.304266 0.952587i \(-0.598411\pi\)
−0.304266 + 0.952587i \(0.598411\pi\)
\(938\) 0 0
\(939\) −8.68629 −0.283466
\(940\) 0 0
\(941\) 10.0000 0.325991 0.162995 0.986627i \(-0.447884\pi\)
0.162995 + 0.986627i \(0.447884\pi\)
\(942\) 0 0
\(943\) −32.5563 −1.06018
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −11.1005 −0.360718 −0.180359 0.983601i \(-0.557726\pi\)
−0.180359 + 0.983601i \(0.557726\pi\)
\(948\) 0 0
\(949\) 7.31371 0.237413
\(950\) 0 0
\(951\) −9.11270 −0.295499
\(952\) 0 0
\(953\) −54.6274 −1.76956 −0.884778 0.466013i \(-0.845690\pi\)
−0.884778 + 0.466013i \(0.845690\pi\)
\(954\) 0 0
\(955\) −5.51472 −0.178452
\(956\) 0 0
\(957\) 2.68629 0.0868355
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −30.3137 −0.977862
\(962\) 0 0
\(963\) −15.7990 −0.509115
\(964\) 0 0
\(965\) 5.31371 0.171054
\(966\) 0 0
\(967\) 24.7574 0.796143 0.398072 0.917354i \(-0.369680\pi\)
0.398072 + 0.917354i \(0.369680\pi\)
\(968\) 0 0
\(969\) −17.9411 −0.576352
\(970\) 0 0
\(971\) 8.00000 0.256732 0.128366 0.991727i \(-0.459027\pi\)
0.128366 + 0.991727i \(0.459027\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −0.828427 −0.0265309
\(976\) 0 0
\(977\) 30.2843 0.968880 0.484440 0.874825i \(-0.339023\pi\)
0.484440 + 0.874825i \(0.339023\pi\)
\(978\) 0 0
\(979\) −4.42641 −0.141469
\(980\) 0 0
\(981\) −51.7990 −1.65381
\(982\) 0 0
\(983\) −41.3848 −1.31997 −0.659985 0.751279i \(-0.729437\pi\)
−0.659985 + 0.751279i \(0.729437\pi\)
\(984\) 0 0
\(985\) −0.343146 −0.0109335
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 38.5391 1.22547
\(990\) 0 0
\(991\) −4.82843 −0.153380 −0.0766900 0.997055i \(-0.524435\pi\)
−0.0766900 + 0.997055i \(0.524435\pi\)
\(992\) 0 0
\(993\) −10.9706 −0.348140
\(994\) 0 0
\(995\) 23.3137 0.739094
\(996\) 0 0
\(997\) 14.3431 0.454252 0.227126 0.973865i \(-0.427067\pi\)
0.227126 + 0.973865i \(0.427067\pi\)
\(998\) 0 0
\(999\) −13.6569 −0.432084
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 3920.2.a.bp.1.2 2
4.3 odd 2 1960.2.a.t.1.1 2
7.3 odd 6 560.2.q.j.401.2 4
7.5 odd 6 560.2.q.j.81.2 4
7.6 odd 2 3920.2.a.bz.1.1 2
20.19 odd 2 9800.2.a.br.1.2 2
28.3 even 6 280.2.q.d.121.1 yes 4
28.11 odd 6 1960.2.q.q.961.2 4
28.19 even 6 280.2.q.d.81.1 4
28.23 odd 6 1960.2.q.q.361.2 4
28.27 even 2 1960.2.a.p.1.2 2
84.47 odd 6 2520.2.bi.k.361.2 4
84.59 odd 6 2520.2.bi.k.1801.2 4
140.3 odd 12 1400.2.bh.g.849.3 8
140.19 even 6 1400.2.q.h.1201.2 4
140.47 odd 12 1400.2.bh.g.249.3 8
140.59 even 6 1400.2.q.h.401.2 4
140.87 odd 12 1400.2.bh.g.849.2 8
140.103 odd 12 1400.2.bh.g.249.2 8
140.139 even 2 9800.2.a.bz.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.q.d.81.1 4 28.19 even 6
280.2.q.d.121.1 yes 4 28.3 even 6
560.2.q.j.81.2 4 7.5 odd 6
560.2.q.j.401.2 4 7.3 odd 6
1400.2.q.h.401.2 4 140.59 even 6
1400.2.q.h.1201.2 4 140.19 even 6
1400.2.bh.g.249.2 8 140.103 odd 12
1400.2.bh.g.249.3 8 140.47 odd 12
1400.2.bh.g.849.2 8 140.87 odd 12
1400.2.bh.g.849.3 8 140.3 odd 12
1960.2.a.p.1.2 2 28.27 even 2
1960.2.a.t.1.1 2 4.3 odd 2
1960.2.q.q.361.2 4 28.23 odd 6
1960.2.q.q.961.2 4 28.11 odd 6
2520.2.bi.k.361.2 4 84.47 odd 6
2520.2.bi.k.1801.2 4 84.59 odd 6
3920.2.a.bp.1.2 2 1.1 even 1 trivial
3920.2.a.bz.1.1 2 7.6 odd 2
9800.2.a.br.1.2 2 20.19 odd 2
9800.2.a.bz.1.1 2 140.139 even 2