Properties

Label 7600.2.a.x.1.2
Level $7600$
Weight $2$
Character 7600.1
Self dual yes
Analytic conductor $60.686$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7600.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.6863055362\)
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 760)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 7600.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.414214 q^{3} -0.414214 q^{7} -2.82843 q^{9} +O(q^{10})\) \(q+0.414214 q^{3} -0.414214 q^{7} -2.82843 q^{9} +1.41421 q^{11} -3.82843 q^{13} +1.00000 q^{17} +1.00000 q^{19} -0.171573 q^{21} +3.24264 q^{23} -2.41421 q^{27} -1.82843 q^{29} -0.585786 q^{31} +0.585786 q^{33} +10.8284 q^{37} -1.58579 q^{39} +7.07107 q^{41} +6.24264 q^{43} -8.00000 q^{47} -6.82843 q^{49} +0.414214 q^{51} +3.82843 q^{53} +0.414214 q^{57} -11.5858 q^{59} +0.585786 q^{61} +1.17157 q^{63} -8.07107 q^{67} +1.34315 q^{69} +11.0711 q^{71} +8.17157 q^{73} -0.585786 q^{77} -4.82843 q^{79} +7.48528 q^{81} -14.4853 q^{83} -0.757359 q^{87} -12.2426 q^{89} +1.58579 q^{91} -0.242641 q^{93} -3.65685 q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 2q^{7} + O(q^{10}) \) \( 2q - 2q^{3} + 2q^{7} - 2q^{13} + 2q^{17} + 2q^{19} - 6q^{21} - 2q^{23} - 2q^{27} + 2q^{29} - 4q^{31} + 4q^{33} + 16q^{37} - 6q^{39} + 4q^{43} - 16q^{47} - 8q^{49} - 2q^{51} + 2q^{53} - 2q^{57} - 26q^{59} + 4q^{61} + 8q^{63} - 2q^{67} + 14q^{69} + 8q^{71} + 22q^{73} - 4q^{77} - 4q^{79} - 2q^{81} - 12q^{83} - 10q^{87} - 16q^{89} + 6q^{91} + 8q^{93} + 4q^{97} - 8q^{99} + O(q^{100}) \)

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
\(5\) 0 0
\(6\) 0 0
\(7\) −0.414214 −0.156558 −0.0782790 0.996931i \(-0.524942\pi\)
−0.0782790 + 0.996931i \(0.524942\pi\)
\(8\) 0 0
\(9\) −2.82843 −0.942809
\(10\) 0 0
\(11\) 1.41421 0.426401 0.213201 0.977008i \(-0.431611\pi\)
0.213201 + 0.977008i \(0.431611\pi\)
\(12\) 0 0
\(13\) −3.82843 −1.06181 −0.530907 0.847430i \(-0.678149\pi\)
−0.530907 + 0.847430i \(0.678149\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.00000 0.242536 0.121268 0.992620i \(-0.461304\pi\)
0.121268 + 0.992620i \(0.461304\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −0.171573 −0.0374403
\(22\) 0 0
\(23\) 3.24264 0.676137 0.338069 0.941121i \(-0.390226\pi\)
0.338069 + 0.941121i \(0.390226\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −2.41421 −0.464616
\(28\) 0 0
\(29\) −1.82843 −0.339530 −0.169765 0.985485i \(-0.554301\pi\)
−0.169765 + 0.985485i \(0.554301\pi\)
\(30\) 0 0
\(31\) −0.585786 −0.105210 −0.0526052 0.998615i \(-0.516752\pi\)
−0.0526052 + 0.998615i \(0.516752\pi\)
\(32\) 0 0
\(33\) 0.585786 0.101972
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 10.8284 1.78018 0.890091 0.455782i \(-0.150640\pi\)
0.890091 + 0.455782i \(0.150640\pi\)
\(38\) 0 0
\(39\) −1.58579 −0.253929
\(40\) 0 0
\(41\) 7.07107 1.10432 0.552158 0.833740i \(-0.313805\pi\)
0.552158 + 0.833740i \(0.313805\pi\)
\(42\) 0 0
\(43\) 6.24264 0.951994 0.475997 0.879447i \(-0.342087\pi\)
0.475997 + 0.879447i \(0.342087\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) −6.82843 −0.975490
\(50\) 0 0
\(51\) 0.414214 0.0580015
\(52\) 0 0
\(53\) 3.82843 0.525875 0.262937 0.964813i \(-0.415309\pi\)
0.262937 + 0.964813i \(0.415309\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0.414214 0.0548639
\(58\) 0 0
\(59\) −11.5858 −1.50834 −0.754170 0.656679i \(-0.771961\pi\)
−0.754170 + 0.656679i \(0.771961\pi\)
\(60\) 0 0
\(61\) 0.585786 0.0750023 0.0375011 0.999297i \(-0.488060\pi\)
0.0375011 + 0.999297i \(0.488060\pi\)
\(62\) 0 0
\(63\) 1.17157 0.147604
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −8.07107 −0.986038 −0.493019 0.870019i \(-0.664107\pi\)
−0.493019 + 0.870019i \(0.664107\pi\)
\(68\) 0 0
\(69\) 1.34315 0.161696
\(70\) 0 0
\(71\) 11.0711 1.31389 0.656947 0.753937i \(-0.271848\pi\)
0.656947 + 0.753937i \(0.271848\pi\)
\(72\) 0 0
\(73\) 8.17157 0.956410 0.478205 0.878248i \(-0.341288\pi\)
0.478205 + 0.878248i \(0.341288\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −0.585786 −0.0667566
\(78\) 0 0
\(79\) −4.82843 −0.543240 −0.271620 0.962405i \(-0.587559\pi\)
−0.271620 + 0.962405i \(0.587559\pi\)
\(80\) 0 0
\(81\) 7.48528 0.831698
\(82\) 0 0
\(83\) −14.4853 −1.58997 −0.794983 0.606632i \(-0.792520\pi\)
−0.794983 + 0.606632i \(0.792520\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −0.757359 −0.0811974
\(88\) 0 0
\(89\) −12.2426 −1.29772 −0.648859 0.760909i \(-0.724753\pi\)
−0.648859 + 0.760909i \(0.724753\pi\)
\(90\) 0 0
\(91\) 1.58579 0.166236
\(92\) 0 0
\(93\) −0.242641 −0.0251607
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −3.65685 −0.371297 −0.185649 0.982616i \(-0.559439\pi\)
−0.185649 + 0.982616i \(0.559439\pi\)
\(98\) 0 0
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) −3.89949 −0.388014 −0.194007 0.981000i \(-0.562148\pi\)
−0.194007 + 0.981000i \(0.562148\pi\)
\(102\) 0 0
\(103\) −9.89949 −0.975426 −0.487713 0.873004i \(-0.662169\pi\)
−0.487713 + 0.873004i \(0.662169\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −0.414214 −0.0400435 −0.0200218 0.999800i \(-0.506374\pi\)
−0.0200218 + 0.999800i \(0.506374\pi\)
\(108\) 0 0
\(109\) −5.00000 −0.478913 −0.239457 0.970907i \(-0.576969\pi\)
−0.239457 + 0.970907i \(0.576969\pi\)
\(110\) 0 0
\(111\) 4.48528 0.425724
\(112\) 0 0
\(113\) 1.07107 0.100758 0.0503788 0.998730i \(-0.483957\pi\)
0.0503788 + 0.998730i \(0.483957\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 10.8284 1.00109
\(118\) 0 0
\(119\) −0.414214 −0.0379709
\(120\) 0 0
\(121\) −9.00000 −0.818182
\(122\) 0 0
\(123\) 2.92893 0.264093
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −16.8284 −1.49328 −0.746641 0.665228i \(-0.768335\pi\)
−0.746641 + 0.665228i \(0.768335\pi\)
\(128\) 0 0
\(129\) 2.58579 0.227666
\(130\) 0 0
\(131\) −10.3431 −0.903685 −0.451842 0.892098i \(-0.649233\pi\)
−0.451842 + 0.892098i \(0.649233\pi\)
\(132\) 0 0
\(133\) −0.414214 −0.0359169
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −6.31371 −0.539417 −0.269708 0.962942i \(-0.586927\pi\)
−0.269708 + 0.962942i \(0.586927\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) −3.31371 −0.279065
\(142\) 0 0
\(143\) −5.41421 −0.452759
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −2.82843 −0.233285
\(148\) 0 0
\(149\) −6.34315 −0.519651 −0.259825 0.965656i \(-0.583665\pi\)
−0.259825 + 0.965656i \(0.583665\pi\)
\(150\) 0 0
\(151\) −16.8284 −1.36948 −0.684739 0.728788i \(-0.740084\pi\)
−0.684739 + 0.728788i \(0.740084\pi\)
\(152\) 0 0
\(153\) −2.82843 −0.228665
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 11.6569 0.930318 0.465159 0.885227i \(-0.345997\pi\)
0.465159 + 0.885227i \(0.345997\pi\)
\(158\) 0 0
\(159\) 1.58579 0.125761
\(160\) 0 0
\(161\) −1.34315 −0.105855
\(162\) 0 0
\(163\) −17.0711 −1.33711 −0.668555 0.743663i \(-0.733087\pi\)
−0.668555 + 0.743663i \(0.733087\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 19.2132 1.48676 0.743381 0.668868i \(-0.233221\pi\)
0.743381 + 0.668868i \(0.233221\pi\)
\(168\) 0 0
\(169\) 1.65685 0.127450
\(170\) 0 0
\(171\) −2.82843 −0.216295
\(172\) 0 0
\(173\) 13.1716 1.00142 0.500708 0.865616i \(-0.333073\pi\)
0.500708 + 0.865616i \(0.333073\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −4.79899 −0.360714
\(178\) 0 0
\(179\) −10.9706 −0.819978 −0.409989 0.912090i \(-0.634468\pi\)
−0.409989 + 0.912090i \(0.634468\pi\)
\(180\) 0 0
\(181\) −16.4853 −1.22534 −0.612671 0.790338i \(-0.709905\pi\)
−0.612671 + 0.790338i \(0.709905\pi\)
\(182\) 0 0
\(183\) 0.242641 0.0179365
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 1.41421 0.103418
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 2.75736 0.199516 0.0997578 0.995012i \(-0.468193\pi\)
0.0997578 + 0.995012i \(0.468193\pi\)
\(192\) 0 0
\(193\) −3.65685 −0.263226 −0.131613 0.991301i \(-0.542016\pi\)
−0.131613 + 0.991301i \(0.542016\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −4.72792 −0.336850 −0.168425 0.985714i \(-0.553868\pi\)
−0.168425 + 0.985714i \(0.553868\pi\)
\(198\) 0 0
\(199\) −1.92893 −0.136738 −0.0683692 0.997660i \(-0.521780\pi\)
−0.0683692 + 0.997660i \(0.521780\pi\)
\(200\) 0 0
\(201\) −3.34315 −0.235807
\(202\) 0 0
\(203\) 0.757359 0.0531562
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −9.17157 −0.637468
\(208\) 0 0
\(209\) 1.41421 0.0978232
\(210\) 0 0
\(211\) 5.10051 0.351133 0.175567 0.984468i \(-0.443824\pi\)
0.175567 + 0.984468i \(0.443824\pi\)
\(212\) 0 0
\(213\) 4.58579 0.314213
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0.242641 0.0164715
\(218\) 0 0
\(219\) 3.38478 0.228722
\(220\) 0 0
\(221\) −3.82843 −0.257528
\(222\) 0 0
\(223\) −8.82843 −0.591195 −0.295598 0.955313i \(-0.595519\pi\)
−0.295598 + 0.955313i \(0.595519\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 8.75736 0.581246 0.290623 0.956838i \(-0.406137\pi\)
0.290623 + 0.956838i \(0.406137\pi\)
\(228\) 0 0
\(229\) 6.97056 0.460628 0.230314 0.973116i \(-0.426025\pi\)
0.230314 + 0.973116i \(0.426025\pi\)
\(230\) 0 0
\(231\) −0.242641 −0.0159646
\(232\) 0 0
\(233\) −13.6569 −0.894690 −0.447345 0.894361i \(-0.647630\pi\)
−0.447345 + 0.894361i \(0.647630\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −2.00000 −0.129914
\(238\) 0 0
\(239\) −27.7279 −1.79357 −0.896785 0.442466i \(-0.854104\pi\)
−0.896785 + 0.442466i \(0.854104\pi\)
\(240\) 0 0
\(241\) −5.65685 −0.364390 −0.182195 0.983262i \(-0.558320\pi\)
−0.182195 + 0.983262i \(0.558320\pi\)
\(242\) 0 0
\(243\) 10.3431 0.663513
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −3.82843 −0.243597
\(248\) 0 0
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) −3.55635 −0.224475 −0.112237 0.993681i \(-0.535802\pi\)
−0.112237 + 0.993681i \(0.535802\pi\)
\(252\) 0 0
\(253\) 4.58579 0.288306
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 28.7279 1.79200 0.895999 0.444056i \(-0.146461\pi\)
0.895999 + 0.444056i \(0.146461\pi\)
\(258\) 0 0
\(259\) −4.48528 −0.278702
\(260\) 0 0
\(261\) 5.17157 0.320112
\(262\) 0 0
\(263\) 15.6569 0.965443 0.482721 0.875774i \(-0.339648\pi\)
0.482721 + 0.875774i \(0.339648\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −5.07107 −0.310344
\(268\) 0 0
\(269\) −10.0000 −0.609711 −0.304855 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) 0 0
\(271\) −15.3848 −0.934559 −0.467279 0.884110i \(-0.654766\pi\)
−0.467279 + 0.884110i \(0.654766\pi\)
\(272\) 0 0
\(273\) 0.656854 0.0397546
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −3.31371 −0.199101 −0.0995507 0.995032i \(-0.531741\pi\)
−0.0995507 + 0.995032i \(0.531741\pi\)
\(278\) 0 0
\(279\) 1.65685 0.0991933
\(280\) 0 0
\(281\) −3.75736 −0.224145 −0.112073 0.993700i \(-0.535749\pi\)
−0.112073 + 0.993700i \(0.535749\pi\)
\(282\) 0 0
\(283\) −9.51472 −0.565591 −0.282796 0.959180i \(-0.591262\pi\)
−0.282796 + 0.959180i \(0.591262\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2.92893 −0.172889
\(288\) 0 0
\(289\) −16.0000 −0.941176
\(290\) 0 0
\(291\) −1.51472 −0.0887944
\(292\) 0 0
\(293\) 10.7990 0.630884 0.315442 0.948945i \(-0.397847\pi\)
0.315442 + 0.948945i \(0.397847\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −3.41421 −0.198113
\(298\) 0 0
\(299\) −12.4142 −0.717933
\(300\) 0 0
\(301\) −2.58579 −0.149042
\(302\) 0 0
\(303\) −1.61522 −0.0927922
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −32.2843 −1.84256 −0.921280 0.388899i \(-0.872855\pi\)
−0.921280 + 0.388899i \(0.872855\pi\)
\(308\) 0 0
\(309\) −4.10051 −0.233270
\(310\) 0 0
\(311\) −23.2426 −1.31797 −0.658985 0.752156i \(-0.729014\pi\)
−0.658985 + 0.752156i \(0.729014\pi\)
\(312\) 0 0
\(313\) 6.65685 0.376268 0.188134 0.982143i \(-0.439756\pi\)
0.188134 + 0.982143i \(0.439756\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 6.79899 0.381869 0.190935 0.981603i \(-0.438848\pi\)
0.190935 + 0.981603i \(0.438848\pi\)
\(318\) 0 0
\(319\) −2.58579 −0.144776
\(320\) 0 0
\(321\) −0.171573 −0.00957626
\(322\) 0 0
\(323\) 1.00000 0.0556415
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −2.07107 −0.114530
\(328\) 0 0
\(329\) 3.31371 0.182691
\(330\) 0 0
\(331\) −19.3848 −1.06548 −0.532742 0.846278i \(-0.678838\pi\)
−0.532742 + 0.846278i \(0.678838\pi\)
\(332\) 0 0
\(333\) −30.6274 −1.67837
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 12.7279 0.693334 0.346667 0.937988i \(-0.387313\pi\)
0.346667 + 0.937988i \(0.387313\pi\)
\(338\) 0 0
\(339\) 0.443651 0.0240958
\(340\) 0 0
\(341\) −0.828427 −0.0448618
\(342\) 0 0
\(343\) 5.72792 0.309279
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −15.1716 −0.814453 −0.407226 0.913327i \(-0.633504\pi\)
−0.407226 + 0.913327i \(0.633504\pi\)
\(348\) 0 0
\(349\) 28.6274 1.53239 0.766195 0.642608i \(-0.222148\pi\)
0.766195 + 0.642608i \(0.222148\pi\)
\(350\) 0 0
\(351\) 9.24264 0.493336
\(352\) 0 0
\(353\) 36.1127 1.92208 0.961042 0.276401i \(-0.0891417\pi\)
0.961042 + 0.276401i \(0.0891417\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −0.171573 −0.00908060
\(358\) 0 0
\(359\) 6.07107 0.320419 0.160209 0.987083i \(-0.448783\pi\)
0.160209 + 0.987083i \(0.448783\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −3.72792 −0.195665
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 33.4558 1.74638 0.873190 0.487379i \(-0.162047\pi\)
0.873190 + 0.487379i \(0.162047\pi\)
\(368\) 0 0
\(369\) −20.0000 −1.04116
\(370\) 0 0
\(371\) −1.58579 −0.0823299
\(372\) 0 0
\(373\) −27.9706 −1.44826 −0.724130 0.689663i \(-0.757759\pi\)
−0.724130 + 0.689663i \(0.757759\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 7.00000 0.360518
\(378\) 0 0
\(379\) 3.38478 0.173864 0.0869321 0.996214i \(-0.472294\pi\)
0.0869321 + 0.996214i \(0.472294\pi\)
\(380\) 0 0
\(381\) −6.97056 −0.357113
\(382\) 0 0
\(383\) 20.7279 1.05915 0.529574 0.848264i \(-0.322352\pi\)
0.529574 + 0.848264i \(0.322352\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −17.6569 −0.897548
\(388\) 0 0
\(389\) 7.89949 0.400520 0.200260 0.979743i \(-0.435821\pi\)
0.200260 + 0.979743i \(0.435821\pi\)
\(390\) 0 0
\(391\) 3.24264 0.163987
\(392\) 0 0
\(393\) −4.28427 −0.216113
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 22.6274 1.13564 0.567819 0.823154i \(-0.307787\pi\)
0.567819 + 0.823154i \(0.307787\pi\)
\(398\) 0 0
\(399\) −0.171573 −0.00858939
\(400\) 0 0
\(401\) −6.10051 −0.304645 −0.152322 0.988331i \(-0.548675\pi\)
−0.152322 + 0.988331i \(0.548675\pi\)
\(402\) 0 0
\(403\) 2.24264 0.111714
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 15.3137 0.759072
\(408\) 0 0
\(409\) 2.58579 0.127859 0.0639295 0.997954i \(-0.479637\pi\)
0.0639295 + 0.997954i \(0.479637\pi\)
\(410\) 0 0
\(411\) −2.61522 −0.128999
\(412\) 0 0
\(413\) 4.79899 0.236143
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 1.65685 0.0811365
\(418\) 0 0
\(419\) 27.2132 1.32945 0.664726 0.747087i \(-0.268548\pi\)
0.664726 + 0.747087i \(0.268548\pi\)
\(420\) 0 0
\(421\) 13.1421 0.640508 0.320254 0.947332i \(-0.396232\pi\)
0.320254 + 0.947332i \(0.396232\pi\)
\(422\) 0 0
\(423\) 22.6274 1.10018
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −0.242641 −0.0117422
\(428\) 0 0
\(429\) −2.24264 −0.108276
\(430\) 0 0
\(431\) 5.41421 0.260793 0.130397 0.991462i \(-0.458375\pi\)
0.130397 + 0.991462i \(0.458375\pi\)
\(432\) 0 0
\(433\) 5.07107 0.243700 0.121850 0.992549i \(-0.461117\pi\)
0.121850 + 0.992549i \(0.461117\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3.24264 0.155117
\(438\) 0 0
\(439\) 30.7279 1.46656 0.733282 0.679925i \(-0.237988\pi\)
0.733282 + 0.679925i \(0.237988\pi\)
\(440\) 0 0
\(441\) 19.3137 0.919700
\(442\) 0 0
\(443\) −15.5563 −0.739104 −0.369552 0.929210i \(-0.620489\pi\)
−0.369552 + 0.929210i \(0.620489\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −2.62742 −0.124273
\(448\) 0 0
\(449\) −35.9411 −1.69617 −0.848083 0.529863i \(-0.822243\pi\)
−0.848083 + 0.529863i \(0.822243\pi\)
\(450\) 0 0
\(451\) 10.0000 0.470882
\(452\) 0 0
\(453\) −6.97056 −0.327506
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 16.3137 0.763123 0.381562 0.924343i \(-0.375386\pi\)
0.381562 + 0.924343i \(0.375386\pi\)
\(458\) 0 0
\(459\) −2.41421 −0.112686
\(460\) 0 0
\(461\) −7.27208 −0.338694 −0.169347 0.985556i \(-0.554166\pi\)
−0.169347 + 0.985556i \(0.554166\pi\)
\(462\) 0 0
\(463\) −22.1421 −1.02903 −0.514516 0.857481i \(-0.672028\pi\)
−0.514516 + 0.857481i \(0.672028\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 21.8995 1.01339 0.506694 0.862126i \(-0.330867\pi\)
0.506694 + 0.862126i \(0.330867\pi\)
\(468\) 0 0
\(469\) 3.34315 0.154372
\(470\) 0 0
\(471\) 4.82843 0.222482
\(472\) 0 0
\(473\) 8.82843 0.405932
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −10.8284 −0.495800
\(478\) 0 0
\(479\) −33.4558 −1.52864 −0.764318 0.644839i \(-0.776924\pi\)
−0.764318 + 0.644839i \(0.776924\pi\)
\(480\) 0 0
\(481\) −41.4558 −1.89022
\(482\) 0 0
\(483\) −0.556349 −0.0253148
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −7.17157 −0.324975 −0.162487 0.986711i \(-0.551952\pi\)
−0.162487 + 0.986711i \(0.551952\pi\)
\(488\) 0 0
\(489\) −7.07107 −0.319765
\(490\) 0 0
\(491\) −17.7574 −0.801378 −0.400689 0.916214i \(-0.631229\pi\)
−0.400689 + 0.916214i \(0.631229\pi\)
\(492\) 0 0
\(493\) −1.82843 −0.0823482
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −4.58579 −0.205701
\(498\) 0 0
\(499\) −18.9289 −0.847375 −0.423688 0.905808i \(-0.639265\pi\)
−0.423688 + 0.905808i \(0.639265\pi\)
\(500\) 0 0
\(501\) 7.95837 0.355554
\(502\) 0 0
\(503\) −32.8995 −1.46692 −0.733458 0.679735i \(-0.762095\pi\)
−0.733458 + 0.679735i \(0.762095\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0.686292 0.0304793
\(508\) 0 0
\(509\) 9.65685 0.428033 0.214016 0.976830i \(-0.431345\pi\)
0.214016 + 0.976830i \(0.431345\pi\)
\(510\) 0 0
\(511\) −3.38478 −0.149734
\(512\) 0 0
\(513\) −2.41421 −0.106590
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −11.3137 −0.497576
\(518\) 0 0
\(519\) 5.45584 0.239485
\(520\) 0 0
\(521\) −28.0000 −1.22670 −0.613351 0.789810i \(-0.710179\pi\)
−0.613351 + 0.789810i \(0.710179\pi\)
\(522\) 0 0
\(523\) 20.2132 0.883862 0.441931 0.897049i \(-0.354294\pi\)
0.441931 + 0.897049i \(0.354294\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −0.585786 −0.0255173
\(528\) 0 0
\(529\) −12.4853 −0.542838
\(530\) 0 0
\(531\) 32.7696 1.42208
\(532\) 0 0
\(533\) −27.0711 −1.17258
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −4.54416 −0.196095
\(538\) 0 0
\(539\) −9.65685 −0.415950
\(540\) 0 0
\(541\) 30.0416 1.29159 0.645795 0.763511i \(-0.276526\pi\)
0.645795 + 0.763511i \(0.276526\pi\)
\(542\) 0 0
\(543\) −6.82843 −0.293036
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 23.9411 1.02365 0.511824 0.859090i \(-0.328970\pi\)
0.511824 + 0.859090i \(0.328970\pi\)
\(548\) 0 0
\(549\) −1.65685 −0.0707128
\(550\) 0 0
\(551\) −1.82843 −0.0778936
\(552\) 0 0
\(553\) 2.00000 0.0850487
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −43.3137 −1.83526 −0.917630 0.397435i \(-0.869900\pi\)
−0.917630 + 0.397435i \(0.869900\pi\)
\(558\) 0 0
\(559\) −23.8995 −1.01084
\(560\) 0 0
\(561\) 0.585786 0.0247319
\(562\) 0 0
\(563\) 44.9706 1.89528 0.947642 0.319336i \(-0.103460\pi\)
0.947642 + 0.319336i \(0.103460\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −3.10051 −0.130209
\(568\) 0 0
\(569\) −8.97056 −0.376066 −0.188033 0.982163i \(-0.560211\pi\)
−0.188033 + 0.982163i \(0.560211\pi\)
\(570\) 0 0
\(571\) −47.6985 −1.99612 −0.998060 0.0622637i \(-0.980168\pi\)
−0.998060 + 0.0622637i \(0.980168\pi\)
\(572\) 0 0
\(573\) 1.14214 0.0477134
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −25.0000 −1.04076 −0.520382 0.853934i \(-0.674210\pi\)
−0.520382 + 0.853934i \(0.674210\pi\)
\(578\) 0 0
\(579\) −1.51472 −0.0629496
\(580\) 0 0
\(581\) 6.00000 0.248922
\(582\) 0 0
\(583\) 5.41421 0.224234
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 32.3848 1.33666 0.668331 0.743864i \(-0.267009\pi\)
0.668331 + 0.743864i \(0.267009\pi\)
\(588\) 0 0
\(589\) −0.585786 −0.0241369
\(590\) 0 0
\(591\) −1.95837 −0.0805566
\(592\) 0 0
\(593\) −42.0000 −1.72473 −0.862367 0.506284i \(-0.831019\pi\)
−0.862367 + 0.506284i \(0.831019\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −0.798990 −0.0327005
\(598\) 0 0
\(599\) 33.5563 1.37108 0.685538 0.728037i \(-0.259567\pi\)
0.685538 + 0.728037i \(0.259567\pi\)
\(600\) 0 0
\(601\) 0.443651 0.0180969 0.00904845 0.999959i \(-0.497120\pi\)
0.00904845 + 0.999959i \(0.497120\pi\)
\(602\) 0 0
\(603\) 22.8284 0.929645
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −30.4853 −1.23736 −0.618680 0.785643i \(-0.712332\pi\)
−0.618680 + 0.785643i \(0.712332\pi\)
\(608\) 0 0
\(609\) 0.313708 0.0127121
\(610\) 0 0
\(611\) 30.6274 1.23905
\(612\) 0 0
\(613\) −2.72792 −0.110180 −0.0550899 0.998481i \(-0.517545\pi\)
−0.0550899 + 0.998481i \(0.517545\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −31.7990 −1.28018 −0.640090 0.768300i \(-0.721103\pi\)
−0.640090 + 0.768300i \(0.721103\pi\)
\(618\) 0 0
\(619\) −6.38478 −0.256626 −0.128313 0.991734i \(-0.540956\pi\)
−0.128313 + 0.991734i \(0.540956\pi\)
\(620\) 0 0
\(621\) −7.82843 −0.314144
\(622\) 0 0
\(623\) 5.07107 0.203168
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0.585786 0.0233941
\(628\) 0 0
\(629\) 10.8284 0.431758
\(630\) 0 0
\(631\) 5.02944 0.200219 0.100109 0.994976i \(-0.468081\pi\)
0.100109 + 0.994976i \(0.468081\pi\)
\(632\) 0 0
\(633\) 2.11270 0.0839722
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 26.1421 1.03579
\(638\) 0 0
\(639\) −31.3137 −1.23875
\(640\) 0 0
\(641\) 3.21320 0.126914 0.0634570 0.997985i \(-0.479787\pi\)
0.0634570 + 0.997985i \(0.479787\pi\)
\(642\) 0 0
\(643\) −36.8284 −1.45237 −0.726186 0.687499i \(-0.758709\pi\)
−0.726186 + 0.687499i \(0.758709\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 2.07107 0.0814221 0.0407110 0.999171i \(-0.487038\pi\)
0.0407110 + 0.999171i \(0.487038\pi\)
\(648\) 0 0
\(649\) −16.3848 −0.643159
\(650\) 0 0
\(651\) 0.100505 0.00393910
\(652\) 0 0
\(653\) 24.2843 0.950317 0.475158 0.879900i \(-0.342391\pi\)
0.475158 + 0.879900i \(0.342391\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −23.1127 −0.901712
\(658\) 0 0
\(659\) −10.8995 −0.424584 −0.212292 0.977206i \(-0.568093\pi\)
−0.212292 + 0.977206i \(0.568093\pi\)
\(660\) 0 0
\(661\) 26.5147 1.03130 0.515652 0.856798i \(-0.327550\pi\)
0.515652 + 0.856798i \(0.327550\pi\)
\(662\) 0 0
\(663\) −1.58579 −0.0615868
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −5.92893 −0.229569
\(668\) 0 0
\(669\) −3.65685 −0.141382
\(670\) 0 0
\(671\) 0.828427 0.0319811
\(672\) 0 0
\(673\) 28.0000 1.07932 0.539660 0.841883i \(-0.318553\pi\)
0.539660 + 0.841883i \(0.318553\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −32.6569 −1.25510 −0.627552 0.778574i \(-0.715943\pi\)
−0.627552 + 0.778574i \(0.715943\pi\)
\(678\) 0 0
\(679\) 1.51472 0.0581296
\(680\) 0 0
\(681\) 3.62742 0.139003
\(682\) 0 0
\(683\) −0.686292 −0.0262602 −0.0131301 0.999914i \(-0.504180\pi\)
−0.0131301 + 0.999914i \(0.504180\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 2.88730 0.110157
\(688\) 0 0
\(689\) −14.6569 −0.558382
\(690\) 0 0
\(691\) 43.4558 1.65314 0.826569 0.562835i \(-0.190289\pi\)
0.826569 + 0.562835i \(0.190289\pi\)
\(692\) 0 0
\(693\) 1.65685 0.0629387
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 7.07107 0.267836
\(698\) 0 0
\(699\) −5.65685 −0.213962
\(700\) 0 0
\(701\) −0.970563 −0.0366576 −0.0183288 0.999832i \(-0.505835\pi\)
−0.0183288 + 0.999832i \(0.505835\pi\)
\(702\) 0 0
\(703\) 10.8284 0.408402
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1.61522 0.0607467
\(708\) 0 0
\(709\) 33.3137 1.25112 0.625561 0.780175i \(-0.284870\pi\)
0.625561 + 0.780175i \(0.284870\pi\)
\(710\) 0 0
\(711\) 13.6569 0.512172
\(712\) 0 0
\(713\) −1.89949 −0.0711366
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −11.4853 −0.428926
\(718\) 0 0
\(719\) −27.5858 −1.02878 −0.514388 0.857557i \(-0.671981\pi\)
−0.514388 + 0.857557i \(0.671981\pi\)
\(720\) 0 0
\(721\) 4.10051 0.152711
\(722\) 0 0
\(723\) −2.34315 −0.0871425
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 44.2132 1.63978 0.819888 0.572523i \(-0.194035\pi\)
0.819888 + 0.572523i \(0.194035\pi\)
\(728\) 0 0
\(729\) −18.1716 −0.673021
\(730\) 0 0
\(731\) 6.24264 0.230892
\(732\) 0 0
\(733\) 34.6274 1.27899 0.639496 0.768794i \(-0.279143\pi\)
0.639496 + 0.768794i \(0.279143\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −11.4142 −0.420448
\(738\) 0 0
\(739\) −4.58579 −0.168691 −0.0843454 0.996437i \(-0.526880\pi\)
−0.0843454 + 0.996437i \(0.526880\pi\)
\(740\) 0 0
\(741\) −1.58579 −0.0582553
\(742\) 0 0
\(743\) −5.75736 −0.211217 −0.105609 0.994408i \(-0.533679\pi\)
−0.105609 + 0.994408i \(0.533679\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 40.9706 1.49903
\(748\) 0 0
\(749\) 0.171573 0.00626914
\(750\) 0 0
\(751\) −38.2426 −1.39549 −0.697747 0.716344i \(-0.745814\pi\)
−0.697747 + 0.716344i \(0.745814\pi\)
\(752\) 0 0
\(753\) −1.47309 −0.0536823
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −27.6569 −1.00521 −0.502603 0.864517i \(-0.667624\pi\)
−0.502603 + 0.864517i \(0.667624\pi\)
\(758\) 0 0
\(759\) 1.89949 0.0689473
\(760\) 0 0
\(761\) −37.9706 −1.37643 −0.688216 0.725506i \(-0.741606\pi\)
−0.688216 + 0.725506i \(0.741606\pi\)
\(762\) 0 0
\(763\) 2.07107 0.0749777
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 44.3553 1.60158
\(768\) 0 0
\(769\) 8.79899 0.317300 0.158650 0.987335i \(-0.449286\pi\)
0.158650 + 0.987335i \(0.449286\pi\)
\(770\) 0 0
\(771\) 11.8995 0.428550
\(772\) 0 0
\(773\) 9.00000 0.323708 0.161854 0.986815i \(-0.448253\pi\)
0.161854 + 0.986815i \(0.448253\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −1.85786 −0.0666505
\(778\) 0 0
\(779\) 7.07107 0.253347
\(780\) 0 0
\(781\) 15.6569 0.560246
\(782\) 0 0
\(783\) 4.41421 0.157751
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 31.2426 1.11368 0.556840 0.830620i \(-0.312014\pi\)
0.556840 + 0.830620i \(0.312014\pi\)
\(788\) 0 0
\(789\) 6.48528 0.230882
\(790\) 0 0
\(791\) −0.443651 −0.0157744
\(792\) 0 0
\(793\) −2.24264 −0.0796385
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 6.85786 0.242918 0.121459 0.992596i \(-0.461243\pi\)
0.121459 + 0.992596i \(0.461243\pi\)
\(798\) 0 0
\(799\) −8.00000 −0.283020
\(800\) 0 0
\(801\) 34.6274 1.22350
\(802\) 0 0
\(803\) 11.5563 0.407815
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −4.14214 −0.145810
\(808\) 0 0
\(809\) −25.4853 −0.896015 −0.448007 0.894030i \(-0.647866\pi\)
−0.448007 + 0.894030i \(0.647866\pi\)
\(810\) 0 0
\(811\) −40.0122 −1.40502 −0.702509 0.711675i \(-0.747937\pi\)
−0.702509 + 0.711675i \(0.747937\pi\)
\(812\) 0 0
\(813\) −6.37258 −0.223496
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 6.24264 0.218402
\(818\) 0 0
\(819\) −4.48528 −0.156728
\(820\) 0 0
\(821\) 27.5980 0.963176 0.481588 0.876398i \(-0.340060\pi\)
0.481588 + 0.876398i \(0.340060\pi\)
\(822\) 0 0
\(823\) −17.5269 −0.610950 −0.305475 0.952200i \(-0.598815\pi\)
−0.305475 + 0.952200i \(0.598815\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −19.8701 −0.690950 −0.345475 0.938428i \(-0.612282\pi\)
−0.345475 + 0.938428i \(0.612282\pi\)
\(828\) 0 0
\(829\) −47.7696 −1.65911 −0.829553 0.558429i \(-0.811404\pi\)
−0.829553 + 0.558429i \(0.811404\pi\)
\(830\) 0 0
\(831\) −1.37258 −0.0476144
\(832\) 0 0
\(833\) −6.82843 −0.236591
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 1.41421 0.0488824
\(838\) 0 0
\(839\) −17.9411 −0.619396 −0.309698 0.950835i \(-0.600228\pi\)
−0.309698 + 0.950835i \(0.600228\pi\)
\(840\) 0 0
\(841\) −25.6569 −0.884719
\(842\) 0 0
\(843\) −1.55635 −0.0536035
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 3.72792 0.128093
\(848\) 0 0
\(849\) −3.94113 −0.135259
\(850\) 0 0
\(851\) 35.1127 1.20365
\(852\) 0 0
\(853\) 47.9411 1.64147 0.820736 0.571307i \(-0.193563\pi\)
0.820736 + 0.571307i \(0.193563\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 16.0000 0.546550 0.273275 0.961936i \(-0.411893\pi\)
0.273275 + 0.961936i \(0.411893\pi\)
\(858\) 0 0
\(859\) 37.2132 1.26970 0.634849 0.772636i \(-0.281062\pi\)
0.634849 + 0.772636i \(0.281062\pi\)
\(860\) 0 0
\(861\) −1.21320 −0.0413459
\(862\) 0 0
\(863\) 20.9289 0.712429 0.356215 0.934404i \(-0.384067\pi\)
0.356215 + 0.934404i \(0.384067\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −6.62742 −0.225079
\(868\) 0 0
\(869\) −6.82843 −0.231639
\(870\) 0 0
\(871\) 30.8995 1.04699
\(872\) 0 0
\(873\) 10.3431 0.350062
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −12.5147 −0.422592 −0.211296 0.977422i \(-0.567768\pi\)
−0.211296 + 0.977422i \(0.567768\pi\)
\(878\) 0 0
\(879\) 4.47309 0.150874
\(880\) 0 0
\(881\) −44.2843 −1.49198 −0.745988 0.665960i \(-0.768022\pi\)
−0.745988 + 0.665960i \(0.768022\pi\)
\(882\) 0 0
\(883\) 35.4558 1.19318 0.596592 0.802545i \(-0.296521\pi\)
0.596592 + 0.802545i \(0.296521\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −15.0711 −0.506037 −0.253018 0.967461i \(-0.581423\pi\)
−0.253018 + 0.967461i \(0.581423\pi\)
\(888\) 0 0
\(889\) 6.97056 0.233785
\(890\) 0 0
\(891\) 10.5858 0.354637
\(892\) 0 0
\(893\) −8.00000 −0.267710
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −5.14214 −0.171691
\(898\) 0 0
\(899\) 1.07107 0.0357221
\(900\) 0 0
\(901\) 3.82843 0.127543
\(902\) 0 0
\(903\) −1.07107 −0.0356429
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 22.5563 0.748971 0.374486 0.927233i \(-0.377819\pi\)
0.374486 + 0.927233i \(0.377819\pi\)
\(908\) 0 0
\(909\) 11.0294 0.365823
\(910\) 0 0
\(911\) −1.65685 −0.0548940 −0.0274470 0.999623i \(-0.508738\pi\)
−0.0274470 + 0.999623i \(0.508738\pi\)
\(912\) 0 0
\(913\) −20.4853 −0.677964
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 4.28427 0.141479
\(918\) 0 0
\(919\) 40.8406 1.34721 0.673604 0.739093i \(-0.264745\pi\)
0.673604 + 0.739093i \(0.264745\pi\)
\(920\) 0 0
\(921\) −13.3726 −0.440642
\(922\) 0 0
\(923\) −42.3848 −1.39511
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 28.0000 0.919641
\(928\) 0 0
\(929\) 8.51472 0.279359 0.139679 0.990197i \(-0.455393\pi\)
0.139679 + 0.990197i \(0.455393\pi\)
\(930\) 0 0
\(931\) −6.82843 −0.223793
\(932\) 0 0
\(933\) −9.62742 −0.315187
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −0.313708 −0.0102484 −0.00512420 0.999987i \(-0.501631\pi\)
−0.00512420 + 0.999987i \(0.501631\pi\)
\(938\) 0 0
\(939\) 2.75736 0.0899830
\(940\) 0 0
\(941\) 28.8579 0.940739 0.470370 0.882469i \(-0.344121\pi\)
0.470370 + 0.882469i \(0.344121\pi\)
\(942\) 0 0
\(943\) 22.9289 0.746669
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −49.2548 −1.60057 −0.800284 0.599622i \(-0.795318\pi\)
−0.800284 + 0.599622i \(0.795318\pi\)
\(948\) 0 0
\(949\) −31.2843 −1.01553
\(950\) 0 0
\(951\) 2.81623 0.0913226
\(952\) 0 0
\(953\) −23.2132 −0.751949 −0.375975 0.926630i \(-0.622692\pi\)
−0.375975 + 0.926630i \(0.622692\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −1.07107 −0.0346227
\(958\) 0 0
\(959\) 2.61522 0.0844500
\(960\) 0 0
\(961\) −30.6569 −0.988931
\(962\) 0 0
\(963\) 1.17157 0.0377534
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −21.8579 −0.702902 −0.351451 0.936206i \(-0.614312\pi\)
−0.351451 + 0.936206i \(0.614312\pi\)
\(968\) 0 0
\(969\) 0.414214 0.0133065
\(970\) 0 0
\(971\) −43.5980 −1.39913 −0.699563 0.714571i \(-0.746622\pi\)
−0.699563 + 0.714571i \(0.746622\pi\)
\(972\) 0 0
\(973\) −1.65685 −0.0531163
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −38.3848 −1.22804 −0.614019 0.789291i \(-0.710448\pi\)
−0.614019 + 0.789291i \(0.710448\pi\)
\(978\) 0 0
\(979\) −17.3137 −0.553349
\(980\) 0 0
\(981\) 14.1421 0.451524
\(982\) 0 0
\(983\) 14.2010 0.452942 0.226471 0.974018i \(-0.427281\pi\)
0.226471 + 0.974018i \(0.427281\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 1.37258 0.0436898
\(988\) 0 0
\(989\) 20.2426 0.643679
\(990\) 0 0
\(991\) 54.5269 1.73210 0.866052 0.499954i \(-0.166650\pi\)
0.866052 + 0.499954i \(0.166650\pi\)
\(992\) 0 0
\(993\) −8.02944 −0.254806
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −3.55635 −0.112631 −0.0563154 0.998413i \(-0.517935\pi\)
−0.0563154 + 0.998413i \(0.517935\pi\)
\(998\) 0 0
\(999\) −26.1421 −0.827101
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 7600.2.a.x.1.2 2
4.3 odd 2 3800.2.a.p.1.1 2
5.2 odd 4 1520.2.d.d.609.2 4
5.3 odd 4 1520.2.d.d.609.3 4
5.4 even 2 7600.2.a.bc.1.1 2
20.3 even 4 760.2.d.c.609.2 4
20.7 even 4 760.2.d.c.609.3 yes 4
20.19 odd 2 3800.2.a.l.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.d.c.609.2 4 20.3 even 4
760.2.d.c.609.3 yes 4 20.7 even 4
1520.2.d.d.609.2 4 5.2 odd 4
1520.2.d.d.609.3 4 5.3 odd 4
3800.2.a.l.1.2 2 20.19 odd 2
3800.2.a.p.1.1 2 4.3 odd 2
7600.2.a.x.1.2 2 1.1 even 1 trivial
7600.2.a.bc.1.1 2 5.4 even 2