Properties

Label 1960.2.a.p.1.2
Level $1960$
Weight $2$
Character 1960.1
Self dual yes
Analytic conductor $15.651$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1960 = 2^{3} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1960.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(15.6506787962\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 1960.1

$q$-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 \(-0.460142\pi\)
0.124890 + 0.992171i \(0.460142\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 0.414214 0.106949
\(16\) 0 0
\(17\) −7.65685 −1.85706 −0.928530 0.371257i \(-0.878927\pi\)
−0.928530 + 0.371257i \(0.878927\pi\)
\(18\) 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 \(-0.697870\pi\)
−0.582358 + 0.812932i \(0.697870\pi\)
\(24\) 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 \(-0.349595\pi\)
0.455124 + 0.890428i \(0.349595\pi\)
\(42\) 0 0
\(43\) −6.89949 −1.05216 −0.526082 0.850434i \(-0.676339\pi\)
−0.526082 + 0.850434i \(0.676339\pi\)
\(44\) 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 \(-0.359866\pi\)
0.426161 + 0.904647i \(0.359866\pi\)
\(62\) 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 \(-0.788864\pi\)
−0.787962 + 0.615724i \(0.788864\pi\)
\(68\) 0 0
\(69\) −2.31371 −0.278538
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) 3.65685 0.428002 0.214001 0.976833i \(-0.431350\pi\)
0.214001 + 0.976833i \(0.431350\pi\)
\(74\) 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 \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 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 \(-0.591391\pi\)
−0.283186 + 0.959065i \(0.591391\pi\)
\(90\) 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 \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 0 0
\(99\) −2.34315 −0.235495
\(100\) 0 0
\(101\) −11.4853 −1.14283 −0.571414 0.820662i \(-0.693605\pi\)
−0.571414 + 0.820662i \(0.693605\pi\)
\(102\) 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 \(-0.587023\pi\)
−0.269999 + 0.962861i \(0.587023\pi\)
\(108\) 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 \(-0.561723\pi\)
−0.192696 + 0.981259i \(0.561723\pi\)
\(128\) 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 \(-0.650205\pi\)
−0.454565 + 0.890714i \(0.650205\pi\)
\(152\) 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 \(-0.483306\pi\)
0.0524227 + 0.998625i \(0.483306\pi\)
\(158\) 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 \(-0.710106\pi\)
−0.613170 + 0.789951i \(0.710106\pi\)
\(164\) 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 \(-0.371372\pi\)
0.393187 + 0.919458i \(0.371372\pi\)
\(174\) 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 \(-0.577924\pi\)
−0.242366 + 0.970185i \(0.577924\pi\)
\(180\) 0 0
\(181\) −4.17157 −0.310071 −0.155035 0.987909i \(-0.549549\pi\)
−0.155035 + 0.987909i \(0.549549\pi\)
\(182\) 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 \(-0.563937\pi\)
−0.199516 + 0.979895i \(0.563937\pi\)
\(192\) 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 \(-0.130959\pi\)
0.916553 + 0.399912i \(0.130959\pi\)
\(212\) 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 \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 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 \(-0.946606\pi\)
−0.985964 + 0.166955i \(0.946606\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 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 \(-0.341499\pi\)
0.477621 + 0.878566i \(0.341499\pi\)
\(258\) 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 \(-0.661149\pi\)
−0.484913 + 0.874563i \(0.661149\pi\)
\(264\) 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 \(-0.565049\pi\)
−0.202938 + 0.979192i \(0.565049\pi\)
\(270\) 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 \(-0.334906\pi\)
0.495715 + 0.868485i \(0.334906\pi\)
\(294\) 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 \(-0.298077\pi\)
0.592663 + 0.805450i \(0.298077\pi\)
\(314\) 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 \(-0.240505\pi\)
0.727881 + 0.685703i \(0.240505\pi\)
\(332\) 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 \(-0.401149\pi\)
0.305583 + 0.952165i \(0.401149\pi\)
\(348\) 0 0
\(349\) 9.82843 0.526104 0.263052 0.964782i \(-0.415271\pi\)
0.263052 + 0.964782i \(0.415271\pi\)
\(350\) 0 0
\(351\) −4.82843 −0.257722
\(352\) 0 0
\(353\) 33.6569 1.79137 0.895687 0.444685i \(-0.146685\pi\)
0.895687 + 0.444685i \(0.146685\pi\)
\(354\) 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 \(-0.554745\pi\)
−0.171140 + 0.985247i \(0.554745\pi\)
\(360\) 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 \(-0.461595\pi\)
0.120359 + 0.992730i \(0.461595\pi\)
\(380\) 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 \(-0.712060\pi\)
−0.618007 + 0.786172i \(0.712060\pi\)
\(398\) 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 \(-0.713517\pi\)
−0.621599 + 0.783335i \(0.713517\pi\)
\(410\) 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 \(-0.506351\pi\)
−0.0199520 + 0.999801i \(0.506351\pi\)
\(432\) 0 0
\(433\) −19.3137 −0.928158 −0.464079 0.885794i \(-0.653615\pi\)
−0.464079 + 0.885794i \(0.653615\pi\)
\(434\) 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 \(-0.620729\pi\)
−0.370252 + 0.928932i \(0.620729\pi\)
\(444\) 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 \(-0.490261\pi\)
0.0305928 + 0.999532i \(0.490261\pi\)
\(462\) 0 0
\(463\) 14.8995 0.692438 0.346219 0.938154i \(-0.387465\pi\)
0.346219 + 0.938154i \(0.387465\pi\)
\(464\) 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 \(-0.834219\pi\)
−0.867413 + 0.497589i \(0.834219\pi\)
\(488\) 0 0
\(489\) −6.48528 −0.293275
\(490\) 0 0
\(491\) 1.51472 0.0683583 0.0341791 0.999416i \(-0.489118\pi\)
0.0341791 + 0.999416i \(0.489118\pi\)
\(492\) 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 \(-0.0492861\pi\)
0.988037 + 0.154219i \(0.0492861\pi\)
\(500\) 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 \(-0.766172\pi\)
−0.742105 + 0.670284i \(0.766172\pi\)
\(510\) 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 \(-0.203589\pi\)
0.802338 + 0.596870i \(0.203589\pi\)
\(522\) 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 \(-0.515468\pi\)
−0.0485735 + 0.998820i \(0.515468\pi\)
\(548\) 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 \(-0.644595\pi\)
−0.438795 + 0.898587i \(0.644595\pi\)
\(572\) 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 \(-0.653533\pi\)
−0.463853 + 0.885912i \(0.653533\pi\)
\(578\) 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 \(-0.289250\pi\)
0.614767 + 0.788709i \(0.289250\pi\)
\(594\) 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 \(-0.163566\pi\)
0.870855 + 0.491541i \(0.163566\pi\)
\(600\) 0 0
\(601\) −34.0000 −1.38689 −0.693444 0.720510i \(-0.743908\pi\)
−0.693444 + 0.720510i \(0.743908\pi\)
\(602\) 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 \(-0.509598\pi\)
−0.0301500 + 0.999545i \(0.509598\pi\)
\(632\) 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 \(-0.420524\pi\)
0.247094 + 0.968992i \(0.420524\pi\)
\(660\) 0 0
\(661\) −50.3137 −1.95698 −0.978488 0.206303i \(-0.933857\pi\)
−0.978488 + 0.206303i \(0.933857\pi\)
\(662\) 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 \(-0.354477\pi\)
0.441415 + 0.897303i \(0.354477\pi\)
\(678\) 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 \(-0.511750\pi\)
−0.0369043 + 0.999319i \(0.511750\pi\)
\(684\) 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 \(-0.454838\pi\)
0.141406 + 0.989952i \(0.454838\pi\)
\(734\) 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 \(-0.399831\pi\)
0.309522 + 0.950892i \(0.399831\pi\)
\(740\) 0 0
\(741\) −4.68629 −0.172155
\(742\) 0 0
\(743\) −10.7574 −0.394649 −0.197325 0.980338i \(-0.563225\pi\)
−0.197325 + 0.980338i \(0.563225\pi\)
\(744\) 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 \(-0.570261\pi\)
−0.218943 + 0.975738i \(0.570261\pi\)
\(752\) 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 \(-0.793282\pi\)
−0.796432 + 0.604728i \(0.793282\pi\)
\(762\) 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 \(-0.215268\pi\)
0.779905 + 0.625898i \(0.215268\pi\)
\(770\) 0 0
\(771\) 6.34315 0.228443
\(772\) 0 0
\(773\) 24.3431 0.875562 0.437781 0.899082i \(-0.355765\pi\)
0.437781 + 0.899082i \(0.355765\pi\)
\(774\) 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 \(-0.715080\pi\)
−0.625438 + 0.780274i \(0.715080\pi\)
\(798\) 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 \(-0.324783\pi\)
0.523080 + 0.852284i \(0.324783\pi\)
\(824\) 0 0
\(825\) 0.343146 0.0119468
\(826\) 0 0
\(827\) 13.5269 0.470377 0.235188 0.971950i \(-0.424429\pi\)
0.235188 + 0.971950i \(0.424429\pi\)
\(828\) 0 0
\(829\) 5.31371 0.184553 0.0922764 0.995733i \(-0.470586\pi\)
0.0922764 + 0.995733i \(0.470586\pi\)
\(830\) 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 \(-0.430312\pi\)
0.217185 + 0.976130i \(0.430312\pi\)
\(854\) 0 0
\(855\) 16.0000 0.547188
\(856\) 0 0
\(857\) −18.9706 −0.648022 −0.324011 0.946053i \(-0.605031\pi\)
−0.324011 + 0.946053i \(0.605031\pi\)
\(858\) 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 \(-0.902378\pi\)
−0.953339 + 0.301903i \(0.902378\pi\)
\(864\) 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 \(-0.647031\pi\)
−0.445660 + 0.895202i \(0.647031\pi\)
\(882\) 0 0
\(883\) 6.68629 0.225012 0.112506 0.993651i \(-0.464112\pi\)
0.112506 + 0.993651i \(0.464112\pi\)
\(884\) 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 \(-0.221628\pi\)
0.767242 + 0.641358i \(0.221628\pi\)
\(908\) 0 0
\(909\) 32.4853 1.07747
\(910\) 0 0
\(911\) −18.4853 −0.612445 −0.306222 0.951960i \(-0.599065\pi\)
−0.306222 + 0.951960i \(0.599065\pi\)
\(912\) 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 \(-0.684796\pi\)
−0.548488 + 0.836159i \(0.684796\pi\)
\(920\) 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 \(-0.451771\pi\)
0.150938 + 0.988543i \(0.451771\pi\)
\(930\) 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 \(-0.401589\pi\)
0.304266 + 0.952587i \(0.401589\pi\)
\(938\) 0 0
\(939\) 8.68629 0.283466
\(940\) 0 0
\(941\) −10.0000 −0.325991 −0.162995 0.986627i \(-0.552116\pi\)
−0.162995 + 0.986627i \(0.552116\pi\)
\(942\) 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 \(-0.442274\pi\)
0.180359 + 0.983601i \(0.442274\pi\)
\(948\) 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 \(-0.630320\pi\)
−0.398072 + 0.917354i \(0.630320\pi\)
\(968\) 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 \(-0.475565\pi\)
0.0766900 + 0.997055i \(0.475565\pi\)
\(992\) 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 \(-0.572933\pi\)
−0.227126 + 0.973865i \(0.572933\pi\)
\(998\) 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 1960.2.a.p.1.2 2
4.3 odd 2 3920.2.a.bz.1.1 2
5.4 even 2 9800.2.a.bz.1.1 2
7.2 even 3 280.2.q.d.81.1 4
7.3 odd 6 1960.2.q.q.961.2 4
7.4 even 3 280.2.q.d.121.1 yes 4
7.5 odd 6 1960.2.q.q.361.2 4
7.6 odd 2 1960.2.a.t.1.1 2
21.2 odd 6 2520.2.bi.k.361.2 4
21.11 odd 6 2520.2.bi.k.1801.2 4
28.11 odd 6 560.2.q.j.401.2 4
28.23 odd 6 560.2.q.j.81.2 4
28.27 even 2 3920.2.a.bp.1.2 2
35.2 odd 12 1400.2.bh.g.249.3 8
35.4 even 6 1400.2.q.h.401.2 4
35.9 even 6 1400.2.q.h.1201.2 4
35.18 odd 12 1400.2.bh.g.849.3 8
35.23 odd 12 1400.2.bh.g.249.2 8
35.32 odd 12 1400.2.bh.g.849.2 8
35.34 odd 2 9800.2.a.br.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.q.d.81.1 4 7.2 even 3
280.2.q.d.121.1 yes 4 7.4 even 3
560.2.q.j.81.2 4 28.23 odd 6
560.2.q.j.401.2 4 28.11 odd 6
1400.2.q.h.401.2 4 35.4 even 6
1400.2.q.h.1201.2 4 35.9 even 6
1400.2.bh.g.249.2 8 35.23 odd 12
1400.2.bh.g.249.3 8 35.2 odd 12
1400.2.bh.g.849.2 8 35.32 odd 12
1400.2.bh.g.849.3 8 35.18 odd 12
1960.2.a.p.1.2 2 1.1 even 1 trivial
1960.2.a.t.1.1 2 7.6 odd 2
1960.2.q.q.361.2 4 7.5 odd 6
1960.2.q.q.961.2 4 7.3 odd 6
2520.2.bi.k.361.2 4 21.2 odd 6
2520.2.bi.k.1801.2 4 21.11 odd 6
3920.2.a.bp.1.2 2 28.27 even 2
3920.2.a.bz.1.1 2 4.3 odd 2
9800.2.a.br.1.2 2 35.34 odd 2
9800.2.a.bz.1.1 2 5.4 even 2