Properties

Label 7800.2.a.bp.1.2
Level $7800$
Weight $2$
Character 7800.1
Self dual yes
Analytic conductor $62.283$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7800.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(62.2833135766\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
Defining polynomial: \(x^{3} - x^{2} - 6 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1560)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.76156\) of defining polynomial
Character \(\chi\) \(=\) 7800.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.00000 q^{9} -2.86464 q^{11} +1.00000 q^{13} -5.52311 q^{17} +3.52311 q^{19} -7.52311 q^{23} +1.00000 q^{27} +6.77551 q^{29} +5.72928 q^{31} -2.86464 q^{33} +3.72928 q^{37} +1.00000 q^{39} -10.1170 q^{41} +5.52311 q^{43} -8.65847 q^{47} -7.00000 q^{49} -5.52311 q^{51} +6.77551 q^{53} +3.52311 q^{57} +0.593923 q^{59} -5.25240 q^{61} +10.5048 q^{67} -7.52311 q^{69} -2.38776 q^{71} -5.45856 q^{73} +2.47689 q^{79} +1.00000 q^{81} -8.11704 q^{83} +6.77551 q^{87} -14.1170 q^{89} +5.72928 q^{93} -6.00000 q^{97} -2.86464 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + 3 q^{9} + O(q^{10}) \) \( 3 q + 3 q^{3} + 3 q^{9} - 6 q^{11} + 3 q^{13} - 4 q^{17} - 2 q^{19} - 10 q^{23} + 3 q^{27} - 10 q^{29} + 12 q^{31} - 6 q^{33} + 6 q^{37} + 3 q^{39} - 10 q^{41} + 4 q^{43} - 16 q^{47} - 21 q^{49} - 4 q^{51} - 10 q^{53} - 2 q^{57} - 6 q^{59} + 2 q^{61} - 4 q^{67} - 10 q^{69} + 8 q^{71} - 6 q^{73} + 20 q^{79} + 3 q^{81} - 4 q^{83} - 10 q^{87} - 22 q^{89} + 12 q^{93} - 18 q^{97} - 6 q^{99} + O(q^{100}) \)

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.86464 −0.863722 −0.431861 0.901940i \(-0.642143\pi\)
−0.431861 + 0.901940i \(0.642143\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.52311 −1.33955 −0.669776 0.742563i \(-0.733610\pi\)
−0.669776 + 0.742563i \(0.733610\pi\)
\(18\) 0 0
\(19\) 3.52311 0.808258 0.404129 0.914702i \(-0.367575\pi\)
0.404129 + 0.914702i \(0.367575\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −7.52311 −1.56868 −0.784339 0.620333i \(-0.786998\pi\)
−0.784339 + 0.620333i \(0.786998\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 6.77551 1.25818 0.629090 0.777332i \(-0.283428\pi\)
0.629090 + 0.777332i \(0.283428\pi\)
\(30\) 0 0
\(31\) 5.72928 1.02901 0.514505 0.857488i \(-0.327976\pi\)
0.514505 + 0.857488i \(0.327976\pi\)
\(32\) 0 0
\(33\) −2.86464 −0.498670
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 3.72928 0.613090 0.306545 0.951856i \(-0.400827\pi\)
0.306545 + 0.951856i \(0.400827\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −10.1170 −1.58002 −0.790008 0.613097i \(-0.789924\pi\)
−0.790008 + 0.613097i \(0.789924\pi\)
\(42\) 0 0
\(43\) 5.52311 0.842267 0.421134 0.906999i \(-0.361632\pi\)
0.421134 + 0.906999i \(0.361632\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −8.65847 −1.26297 −0.631484 0.775389i \(-0.717554\pi\)
−0.631484 + 0.775389i \(0.717554\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) −5.52311 −0.773391
\(52\) 0 0
\(53\) 6.77551 0.930688 0.465344 0.885130i \(-0.345931\pi\)
0.465344 + 0.885130i \(0.345931\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 3.52311 0.466648
\(58\) 0 0
\(59\) 0.593923 0.0773221 0.0386611 0.999252i \(-0.487691\pi\)
0.0386611 + 0.999252i \(0.487691\pi\)
\(60\) 0 0
\(61\) −5.25240 −0.672500 −0.336250 0.941773i \(-0.609159\pi\)
−0.336250 + 0.941773i \(0.609159\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 10.5048 1.28336 0.641682 0.766971i \(-0.278237\pi\)
0.641682 + 0.766971i \(0.278237\pi\)
\(68\) 0 0
\(69\) −7.52311 −0.905677
\(70\) 0 0
\(71\) −2.38776 −0.283374 −0.141687 0.989911i \(-0.545253\pi\)
−0.141687 + 0.989911i \(0.545253\pi\)
\(72\) 0 0
\(73\) −5.45856 −0.638877 −0.319438 0.947607i \(-0.603494\pi\)
−0.319438 + 0.947607i \(0.603494\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 2.47689 0.278671 0.139336 0.990245i \(-0.455503\pi\)
0.139336 + 0.990245i \(0.455503\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −8.11704 −0.890961 −0.445480 0.895292i \(-0.646967\pi\)
−0.445480 + 0.895292i \(0.646967\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 6.77551 0.726411
\(88\) 0 0
\(89\) −14.1170 −1.49640 −0.748201 0.663472i \(-0.769082\pi\)
−0.748201 + 0.663472i \(0.769082\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 5.72928 0.594099
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −6.00000 −0.609208 −0.304604 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 0 0
\(99\) −2.86464 −0.287907
\(100\) 0 0
\(101\) −11.7293 −1.16711 −0.583554 0.812075i \(-0.698338\pi\)
−0.583554 + 0.812075i \(0.698338\pi\)
\(102\) 0 0
\(103\) −14.7755 −1.45587 −0.727937 0.685644i \(-0.759521\pi\)
−0.727937 + 0.685644i \(0.759521\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1.72928 −0.167176 −0.0835880 0.996500i \(-0.526638\pi\)
−0.0835880 + 0.996500i \(0.526638\pi\)
\(108\) 0 0
\(109\) 12.0279 1.15206 0.576032 0.817427i \(-0.304600\pi\)
0.576032 + 0.817427i \(0.304600\pi\)
\(110\) 0 0
\(111\) 3.72928 0.353968
\(112\) 0 0
\(113\) 7.25240 0.682248 0.341124 0.940018i \(-0.389192\pi\)
0.341124 + 0.940018i \(0.389192\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −2.79383 −0.253985
\(122\) 0 0
\(123\) −10.1170 −0.912223
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −17.2524 −1.53090 −0.765451 0.643494i \(-0.777484\pi\)
−0.765451 + 0.643494i \(0.777484\pi\)
\(128\) 0 0
\(129\) 5.52311 0.486283
\(130\) 0 0
\(131\) −8.77551 −0.766720 −0.383360 0.923599i \(-0.625233\pi\)
−0.383360 + 0.923599i \(0.625233\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −19.4340 −1.66036 −0.830179 0.557497i \(-0.811762\pi\)
−0.830179 + 0.557497i \(0.811762\pi\)
\(138\) 0 0
\(139\) 7.58767 0.643577 0.321789 0.946812i \(-0.395716\pi\)
0.321789 + 0.946812i \(0.395716\pi\)
\(140\) 0 0
\(141\) −8.65847 −0.729175
\(142\) 0 0
\(143\) −2.86464 −0.239553
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −7.00000 −0.577350
\(148\) 0 0
\(149\) −1.40608 −0.115190 −0.0575952 0.998340i \(-0.518343\pi\)
−0.0575952 + 0.998340i \(0.518343\pi\)
\(150\) 0 0
\(151\) −1.31695 −0.107172 −0.0535858 0.998563i \(-0.517065\pi\)
−0.0535858 + 0.998563i \(0.517065\pi\)
\(152\) 0 0
\(153\) −5.52311 −0.446517
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −20.7110 −1.65291 −0.826457 0.562999i \(-0.809647\pi\)
−0.826457 + 0.562999i \(0.809647\pi\)
\(158\) 0 0
\(159\) 6.77551 0.537333
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −2.27072 −0.177856 −0.0889282 0.996038i \(-0.528344\pi\)
−0.0889282 + 0.996038i \(0.528344\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −7.34153 −0.568104 −0.284052 0.958809i \(-0.591679\pi\)
−0.284052 + 0.958809i \(0.591679\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 3.52311 0.269419
\(172\) 0 0
\(173\) 16.0925 1.22349 0.611743 0.791056i \(-0.290468\pi\)
0.611743 + 0.791056i \(0.290468\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0.593923 0.0446420
\(178\) 0 0
\(179\) 15.0462 1.12461 0.562304 0.826931i \(-0.309915\pi\)
0.562304 + 0.826931i \(0.309915\pi\)
\(180\) 0 0
\(181\) 5.58767 0.415328 0.207664 0.978200i \(-0.433414\pi\)
0.207664 + 0.978200i \(0.433414\pi\)
\(182\) 0 0
\(183\) −5.25240 −0.388268
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 15.8217 1.15700
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 17.5510 1.26995 0.634974 0.772534i \(-0.281011\pi\)
0.634974 + 0.772534i \(0.281011\pi\)
\(192\) 0 0
\(193\) −13.8217 −0.994911 −0.497455 0.867490i \(-0.665732\pi\)
−0.497455 + 0.867490i \(0.665732\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −26.3511 −1.87744 −0.938719 0.344682i \(-0.887987\pi\)
−0.938719 + 0.344682i \(0.887987\pi\)
\(198\) 0 0
\(199\) −23.0741 −1.63568 −0.817841 0.575444i \(-0.804829\pi\)
−0.817841 + 0.575444i \(0.804829\pi\)
\(200\) 0 0
\(201\) 10.5048 0.740951
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −7.52311 −0.522893
\(208\) 0 0
\(209\) −10.0925 −0.698110
\(210\) 0 0
\(211\) 1.52311 0.104856 0.0524278 0.998625i \(-0.483304\pi\)
0.0524278 + 0.998625i \(0.483304\pi\)
\(212\) 0 0
\(213\) −2.38776 −0.163606
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −5.45856 −0.368856
\(220\) 0 0
\(221\) −5.52311 −0.371525
\(222\) 0 0
\(223\) −13.3169 −0.891769 −0.445884 0.895091i \(-0.647111\pi\)
−0.445884 + 0.895091i \(0.647111\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 6.80009 0.451338 0.225669 0.974204i \(-0.427543\pi\)
0.225669 + 0.974204i \(0.427543\pi\)
\(228\) 0 0
\(229\) 1.52311 0.100650 0.0503251 0.998733i \(-0.483974\pi\)
0.0503251 + 0.998733i \(0.483974\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 10.7110 0.701698 0.350849 0.936432i \(-0.385893\pi\)
0.350849 + 0.936432i \(0.385893\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 2.47689 0.160891
\(238\) 0 0
\(239\) 4.65847 0.301332 0.150666 0.988585i \(-0.451858\pi\)
0.150666 + 0.988585i \(0.451858\pi\)
\(240\) 0 0
\(241\) 5.22449 0.336539 0.168269 0.985741i \(-0.446182\pi\)
0.168269 + 0.985741i \(0.446182\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 3.52311 0.224170
\(248\) 0 0
\(249\) −8.11704 −0.514396
\(250\) 0 0
\(251\) 5.85838 0.369778 0.184889 0.982759i \(-0.440807\pi\)
0.184889 + 0.982759i \(0.440807\pi\)
\(252\) 0 0
\(253\) 21.5510 1.35490
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2.29862 0.143384 0.0716921 0.997427i \(-0.477160\pi\)
0.0716921 + 0.997427i \(0.477160\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 6.77551 0.419394
\(262\) 0 0
\(263\) 31.5789 1.94724 0.973620 0.228175i \(-0.0732760\pi\)
0.973620 + 0.228175i \(0.0732760\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −14.1170 −0.863949
\(268\) 0 0
\(269\) −30.3632 −1.85128 −0.925638 0.378411i \(-0.876471\pi\)
−0.925638 + 0.378411i \(0.876471\pi\)
\(270\) 0 0
\(271\) 28.5972 1.73716 0.868580 0.495550i \(-0.165033\pi\)
0.868580 + 0.495550i \(0.165033\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −1.79383 −0.107781 −0.0538905 0.998547i \(-0.517162\pi\)
−0.0538905 + 0.998547i \(0.517162\pi\)
\(278\) 0 0
\(279\) 5.72928 0.343003
\(280\) 0 0
\(281\) 32.2095 1.92146 0.960729 0.277489i \(-0.0895024\pi\)
0.960729 + 0.277489i \(0.0895024\pi\)
\(282\) 0 0
\(283\) −18.0925 −1.07548 −0.537742 0.843109i \(-0.680723\pi\)
−0.537742 + 0.843109i \(0.680723\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 13.5048 0.794400
\(290\) 0 0
\(291\) −6.00000 −0.351726
\(292\) 0 0
\(293\) 12.6218 0.737375 0.368688 0.929553i \(-0.379807\pi\)
0.368688 + 0.929553i \(0.379807\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −2.86464 −0.166223
\(298\) 0 0
\(299\) −7.52311 −0.435073
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −11.7293 −0.673830
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −20.0000 −1.14146 −0.570730 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) 0 0
\(309\) −14.7755 −0.840549
\(310\) 0 0
\(311\) −31.8217 −1.80445 −0.902223 0.431271i \(-0.858065\pi\)
−0.902223 + 0.431271i \(0.858065\pi\)
\(312\) 0 0
\(313\) −11.7938 −0.666627 −0.333313 0.942816i \(-0.608167\pi\)
−0.333313 + 0.942816i \(0.608167\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −11.4340 −0.642197 −0.321098 0.947046i \(-0.604052\pi\)
−0.321098 + 0.947046i \(0.604052\pi\)
\(318\) 0 0
\(319\) −19.4094 −1.08672
\(320\) 0 0
\(321\) −1.72928 −0.0965191
\(322\) 0 0
\(323\) −19.4586 −1.08270
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 12.0279 0.665145
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 2.20617 0.121262 0.0606310 0.998160i \(-0.480689\pi\)
0.0606310 + 0.998160i \(0.480689\pi\)
\(332\) 0 0
\(333\) 3.72928 0.204363
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 16.8034 0.915340 0.457670 0.889122i \(-0.348684\pi\)
0.457670 + 0.889122i \(0.348684\pi\)
\(338\) 0 0
\(339\) 7.25240 0.393896
\(340\) 0 0
\(341\) −16.4123 −0.888778
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −9.96336 −0.534861 −0.267430 0.963577i \(-0.586175\pi\)
−0.267430 + 0.963577i \(0.586175\pi\)
\(348\) 0 0
\(349\) −21.3449 −1.14256 −0.571282 0.820754i \(-0.693554\pi\)
−0.571282 + 0.820754i \(0.693554\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) −33.3973 −1.77756 −0.888781 0.458333i \(-0.848447\pi\)
−0.888781 + 0.458333i \(0.848447\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 21.9754 1.15982 0.579909 0.814681i \(-0.303088\pi\)
0.579909 + 0.814681i \(0.303088\pi\)
\(360\) 0 0
\(361\) −6.58767 −0.346719
\(362\) 0 0
\(363\) −2.79383 −0.146638
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −29.2803 −1.52842 −0.764210 0.644968i \(-0.776871\pi\)
−0.764210 + 0.644968i \(0.776871\pi\)
\(368\) 0 0
\(369\) −10.1170 −0.526672
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −2.41233 −0.124906 −0.0624530 0.998048i \(-0.519892\pi\)
−0.0624530 + 0.998048i \(0.519892\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 6.77551 0.348957
\(378\) 0 0
\(379\) 19.5231 1.00284 0.501418 0.865205i \(-0.332812\pi\)
0.501418 + 0.865205i \(0.332812\pi\)
\(380\) 0 0
\(381\) −17.2524 −0.883867
\(382\) 0 0
\(383\) 23.7047 1.21125 0.605627 0.795749i \(-0.292922\pi\)
0.605627 + 0.795749i \(0.292922\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 5.52311 0.280756
\(388\) 0 0
\(389\) −14.0000 −0.709828 −0.354914 0.934899i \(-0.615490\pi\)
−0.354914 + 0.934899i \(0.615490\pi\)
\(390\) 0 0
\(391\) 41.5510 2.10133
\(392\) 0 0
\(393\) −8.77551 −0.442666
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −28.8680 −1.44884 −0.724421 0.689358i \(-0.757893\pi\)
−0.724421 + 0.689358i \(0.757893\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 20.0804 1.00277 0.501383 0.865225i \(-0.332825\pi\)
0.501383 + 0.865225i \(0.332825\pi\)
\(402\) 0 0
\(403\) 5.72928 0.285396
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −10.6831 −0.529539
\(408\) 0 0
\(409\) −13.6926 −0.677057 −0.338529 0.940956i \(-0.609929\pi\)
−0.338529 + 0.940956i \(0.609929\pi\)
\(410\) 0 0
\(411\) −19.4340 −0.958608
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 7.58767 0.371570
\(418\) 0 0
\(419\) −16.3632 −0.799393 −0.399697 0.916647i \(-0.630885\pi\)
−0.399697 + 0.916647i \(0.630885\pi\)
\(420\) 0 0
\(421\) 24.9817 1.21753 0.608766 0.793350i \(-0.291665\pi\)
0.608766 + 0.793350i \(0.291665\pi\)
\(422\) 0 0
\(423\) −8.65847 −0.420989
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −2.86464 −0.138306
\(430\) 0 0
\(431\) −27.9388 −1.34576 −0.672882 0.739750i \(-0.734944\pi\)
−0.672882 + 0.739750i \(0.734944\pi\)
\(432\) 0 0
\(433\) 26.7110 1.28365 0.641823 0.766852i \(-0.278178\pi\)
0.641823 + 0.766852i \(0.278178\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −26.5048 −1.26790
\(438\) 0 0
\(439\) −5.00958 −0.239094 −0.119547 0.992829i \(-0.538144\pi\)
−0.119547 + 0.992829i \(0.538144\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) 0 0
\(443\) 19.6926 0.935625 0.467813 0.883828i \(-0.345042\pi\)
0.467813 + 0.883828i \(0.345042\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −1.40608 −0.0665052
\(448\) 0 0
\(449\) 38.7143 1.82704 0.913520 0.406794i \(-0.133353\pi\)
0.913520 + 0.406794i \(0.133353\pi\)
\(450\) 0 0
\(451\) 28.9817 1.36469
\(452\) 0 0
\(453\) −1.31695 −0.0618756
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −10.9538 −0.512396 −0.256198 0.966624i \(-0.582470\pi\)
−0.256198 + 0.966624i \(0.582470\pi\)
\(458\) 0 0
\(459\) −5.52311 −0.257797
\(460\) 0 0
\(461\) −0.864641 −0.0402703 −0.0201352 0.999797i \(-0.506410\pi\)
−0.0201352 + 0.999797i \(0.506410\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −20.7755 −0.961376 −0.480688 0.876892i \(-0.659613\pi\)
−0.480688 + 0.876892i \(0.659613\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −20.7110 −0.954311
\(472\) 0 0
\(473\) −15.8217 −0.727484
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 6.77551 0.310229
\(478\) 0 0
\(479\) 20.6585 0.943910 0.471955 0.881623i \(-0.343549\pi\)
0.471955 + 0.881623i \(0.343549\pi\)
\(480\) 0 0
\(481\) 3.72928 0.170041
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 16.3632 0.741486 0.370743 0.928735i \(-0.379103\pi\)
0.370743 + 0.928735i \(0.379103\pi\)
\(488\) 0 0
\(489\) −2.27072 −0.102685
\(490\) 0 0
\(491\) 26.7389 1.20671 0.603354 0.797473i \(-0.293831\pi\)
0.603354 + 0.797473i \(0.293831\pi\)
\(492\) 0 0
\(493\) −37.4219 −1.68540
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 4.24281 0.189934 0.0949672 0.995480i \(-0.469725\pi\)
0.0949672 + 0.995480i \(0.469725\pi\)
\(500\) 0 0
\(501\) −7.34153 −0.327995
\(502\) 0 0
\(503\) 0.889220 0.0396484 0.0198242 0.999803i \(-0.493689\pi\)
0.0198242 + 0.999803i \(0.493689\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 1.00000 0.0444116
\(508\) 0 0
\(509\) −11.3694 −0.503941 −0.251971 0.967735i \(-0.581079\pi\)
−0.251971 + 0.967735i \(0.581079\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 3.52311 0.155549
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 24.8034 1.09085
\(518\) 0 0
\(519\) 16.0925 0.706380
\(520\) 0 0
\(521\) 3.18785 0.139662 0.0698310 0.997559i \(-0.477754\pi\)
0.0698310 + 0.997559i \(0.477754\pi\)
\(522\) 0 0
\(523\) 37.1666 1.62518 0.812591 0.582835i \(-0.198056\pi\)
0.812591 + 0.582835i \(0.198056\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −31.6435 −1.37841
\(528\) 0 0
\(529\) 33.5972 1.46075
\(530\) 0 0
\(531\) 0.593923 0.0257740
\(532\) 0 0
\(533\) −10.1170 −0.438218
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 15.0462 0.649293
\(538\) 0 0
\(539\) 20.0525 0.863722
\(540\) 0 0
\(541\) −8.02791 −0.345147 −0.172573 0.984997i \(-0.555208\pi\)
−0.172573 + 0.984997i \(0.555208\pi\)
\(542\) 0 0
\(543\) 5.58767 0.239790
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −31.2032 −1.33415 −0.667077 0.744989i \(-0.732455\pi\)
−0.667077 + 0.744989i \(0.732455\pi\)
\(548\) 0 0
\(549\) −5.25240 −0.224167
\(550\) 0 0
\(551\) 23.8709 1.01693
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 29.2191 1.23805 0.619026 0.785370i \(-0.287528\pi\)
0.619026 + 0.785370i \(0.287528\pi\)
\(558\) 0 0
\(559\) 5.52311 0.233603
\(560\) 0 0
\(561\) 15.8217 0.667994
\(562\) 0 0
\(563\) −30.5048 −1.28562 −0.642812 0.766024i \(-0.722232\pi\)
−0.642812 + 0.766024i \(0.722232\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −12.1416 −0.509003 −0.254502 0.967072i \(-0.581911\pi\)
−0.254502 + 0.967072i \(0.581911\pi\)
\(570\) 0 0
\(571\) −8.85258 −0.370469 −0.185234 0.982694i \(-0.559304\pi\)
−0.185234 + 0.982694i \(0.559304\pi\)
\(572\) 0 0
\(573\) 17.5510 0.733204
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 30.2341 1.25866 0.629330 0.777138i \(-0.283329\pi\)
0.629330 + 0.777138i \(0.283329\pi\)
\(578\) 0 0
\(579\) −13.8217 −0.574412
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −19.4094 −0.803855
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 23.0342 0.950722 0.475361 0.879791i \(-0.342318\pi\)
0.475361 + 0.879791i \(0.342318\pi\)
\(588\) 0 0
\(589\) 20.1849 0.831705
\(590\) 0 0
\(591\) −26.3511 −1.08394
\(592\) 0 0
\(593\) −27.9754 −1.14881 −0.574406 0.818570i \(-0.694767\pi\)
−0.574406 + 0.818570i \(0.694767\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −23.0741 −0.944361
\(598\) 0 0
\(599\) 22.7389 0.929085 0.464542 0.885551i \(-0.346219\pi\)
0.464542 + 0.885551i \(0.346219\pi\)
\(600\) 0 0
\(601\) −11.2158 −0.457500 −0.228750 0.973485i \(-0.573464\pi\)
−0.228750 + 0.973485i \(0.573464\pi\)
\(602\) 0 0
\(603\) 10.5048 0.427788
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −23.7293 −0.963142 −0.481571 0.876407i \(-0.659934\pi\)
−0.481571 + 0.876407i \(0.659934\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −8.65847 −0.350284
\(612\) 0 0
\(613\) 18.2341 0.736467 0.368234 0.929733i \(-0.379963\pi\)
0.368234 + 0.929733i \(0.379963\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −0.0245796 −0.000989536 0 −0.000494768 1.00000i \(-0.500157\pi\)
−0.000494768 1.00000i \(0.500157\pi\)
\(618\) 0 0
\(619\) 31.2158 1.25467 0.627334 0.778751i \(-0.284146\pi\)
0.627334 + 0.778751i \(0.284146\pi\)
\(620\) 0 0
\(621\) −7.52311 −0.301892
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −10.0925 −0.403054
\(628\) 0 0
\(629\) −20.5972 −0.821266
\(630\) 0 0
\(631\) 13.4460 0.535279 0.267639 0.963519i \(-0.413756\pi\)
0.267639 + 0.963519i \(0.413756\pi\)
\(632\) 0 0
\(633\) 1.52311 0.0605384
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −7.00000 −0.277350
\(638\) 0 0
\(639\) −2.38776 −0.0944581
\(640\) 0 0
\(641\) 2.54144 0.100381 0.0501904 0.998740i \(-0.484017\pi\)
0.0501904 + 0.998740i \(0.484017\pi\)
\(642\) 0 0
\(643\) −13.9634 −0.550661 −0.275330 0.961350i \(-0.588787\pi\)
−0.275330 + 0.961350i \(0.588787\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 30.8034 1.21101 0.605504 0.795843i \(-0.292972\pi\)
0.605504 + 0.795843i \(0.292972\pi\)
\(648\) 0 0
\(649\) −1.70138 −0.0667848
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −2.00000 −0.0782660 −0.0391330 0.999234i \(-0.512460\pi\)
−0.0391330 + 0.999234i \(0.512460\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −5.45856 −0.212959
\(658\) 0 0
\(659\) −4.05581 −0.157992 −0.0789960 0.996875i \(-0.525171\pi\)
−0.0789960 + 0.996875i \(0.525171\pi\)
\(660\) 0 0
\(661\) −32.9325 −1.28093 −0.640463 0.767989i \(-0.721258\pi\)
−0.640463 + 0.767989i \(0.721258\pi\)
\(662\) 0 0
\(663\) −5.52311 −0.214500
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −50.9729 −1.97368
\(668\) 0 0
\(669\) −13.3169 −0.514863
\(670\) 0 0
\(671\) 15.0462 0.580853
\(672\) 0 0
\(673\) −9.42192 −0.363188 −0.181594 0.983374i \(-0.558126\pi\)
−0.181594 + 0.983374i \(0.558126\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −42.2341 −1.62319 −0.811594 0.584222i \(-0.801400\pi\)
−0.811594 + 0.584222i \(0.801400\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 6.80009 0.260580
\(682\) 0 0
\(683\) 18.7509 0.717484 0.358742 0.933437i \(-0.383206\pi\)
0.358742 + 0.933437i \(0.383206\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 1.52311 0.0581104
\(688\) 0 0
\(689\) 6.77551 0.258126
\(690\) 0 0
\(691\) 45.2524 1.72148 0.860741 0.509043i \(-0.170001\pi\)
0.860741 + 0.509043i \(0.170001\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 55.8776 2.11651
\(698\) 0 0
\(699\) 10.7110 0.405126
\(700\) 0 0
\(701\) −37.4586 −1.41479 −0.707395 0.706818i \(-0.750130\pi\)
−0.707395 + 0.706818i \(0.750130\pi\)
\(702\) 0 0
\(703\) 13.1387 0.495535
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 11.8988 0.446869 0.223435 0.974719i \(-0.428273\pi\)
0.223435 + 0.974719i \(0.428273\pi\)
\(710\) 0 0
\(711\) 2.47689 0.0928905
\(712\) 0 0
\(713\) −43.1020 −1.61418
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 4.65847 0.173974
\(718\) 0 0
\(719\) 25.7851 0.961622 0.480811 0.876824i \(-0.340342\pi\)
0.480811 + 0.876824i \(0.340342\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 5.22449 0.194301
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −34.8034 −1.29079 −0.645394 0.763850i \(-0.723307\pi\)
−0.645394 + 0.763850i \(0.723307\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −30.5048 −1.12826
\(732\) 0 0
\(733\) −19.5510 −0.722133 −0.361067 0.932540i \(-0.617587\pi\)
−0.361067 + 0.932540i \(0.617587\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −30.0925 −1.10847
\(738\) 0 0
\(739\) 12.2986 0.452412 0.226206 0.974079i \(-0.427368\pi\)
0.226206 + 0.974079i \(0.427368\pi\)
\(740\) 0 0
\(741\) 3.52311 0.129425
\(742\) 0 0
\(743\) −40.9975 −1.50405 −0.752027 0.659133i \(-0.770924\pi\)
−0.752027 + 0.659133i \(0.770924\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −8.11704 −0.296987
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 33.5510 1.22429 0.612147 0.790744i \(-0.290306\pi\)
0.612147 + 0.790744i \(0.290306\pi\)
\(752\) 0 0
\(753\) 5.85838 0.213491
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −25.1020 −0.912349 −0.456175 0.889890i \(-0.650781\pi\)
−0.456175 + 0.889890i \(0.650781\pi\)
\(758\) 0 0
\(759\) 21.5510 0.782253
\(760\) 0 0
\(761\) −39.9021 −1.44645 −0.723226 0.690612i \(-0.757341\pi\)
−0.723226 + 0.690612i \(0.757341\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0.593923 0.0214453
\(768\) 0 0
\(769\) −15.5510 −0.560784 −0.280392 0.959886i \(-0.590464\pi\)
−0.280392 + 0.959886i \(0.590464\pi\)
\(770\) 0 0
\(771\) 2.29862 0.0827830
\(772\) 0 0
\(773\) −27.7972 −0.999794 −0.499897 0.866085i \(-0.666629\pi\)
−0.499897 + 0.866085i \(0.666629\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −35.6435 −1.27706
\(780\) 0 0
\(781\) 6.84006 0.244757
\(782\) 0 0
\(783\) 6.77551 0.242137
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 30.6339 1.09198 0.545990 0.837792i \(-0.316154\pi\)
0.545990 + 0.837792i \(0.316154\pi\)
\(788\) 0 0
\(789\) 31.5789 1.12424
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −5.25240 −0.186518
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 18.6464 0.660490 0.330245 0.943895i \(-0.392869\pi\)
0.330245 + 0.943895i \(0.392869\pi\)
\(798\) 0 0
\(799\) 47.8217 1.69181
\(800\) 0 0
\(801\) −14.1170 −0.498801
\(802\) 0 0
\(803\) 15.6368 0.551812
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −30.3632 −1.06883
\(808\) 0 0
\(809\) −54.8313 −1.92777 −0.963883 0.266325i \(-0.914191\pi\)
−0.963883 + 0.266325i \(0.914191\pi\)
\(810\) 0 0
\(811\) 5.97209 0.209709 0.104854 0.994488i \(-0.466562\pi\)
0.104854 + 0.994488i \(0.466562\pi\)
\(812\) 0 0
\(813\) 28.5972 1.00295
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 19.4586 0.680769
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 30.2866 1.05701 0.528504 0.848931i \(-0.322753\pi\)
0.528504 + 0.848931i \(0.322753\pi\)
\(822\) 0 0
\(823\) 31.2437 1.08909 0.544543 0.838733i \(-0.316703\pi\)
0.544543 + 0.838733i \(0.316703\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 21.9754 0.764160 0.382080 0.924129i \(-0.375208\pi\)
0.382080 + 0.924129i \(0.375208\pi\)
\(828\) 0 0
\(829\) −11.7014 −0.406406 −0.203203 0.979137i \(-0.565135\pi\)
−0.203203 + 0.979137i \(0.565135\pi\)
\(830\) 0 0
\(831\) −1.79383 −0.0622274
\(832\) 0 0
\(833\) 38.6618 1.33955
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 5.72928 0.198033
\(838\) 0 0
\(839\) 11.7047 0.404091 0.202046 0.979376i \(-0.435241\pi\)
0.202046 + 0.979376i \(0.435241\pi\)
\(840\) 0 0
\(841\) 16.9075 0.583019
\(842\) 0 0
\(843\) 32.2095 1.10935
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −18.0925 −0.620932
\(850\) 0 0
\(851\) −28.0558 −0.961741
\(852\) 0 0
\(853\) 15.3169 0.524442 0.262221 0.965008i \(-0.415545\pi\)
0.262221 + 0.965008i \(0.415545\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −20.7476 −0.708725 −0.354362 0.935108i \(-0.615302\pi\)
−0.354362 + 0.935108i \(0.615302\pi\)
\(858\) 0 0
\(859\) 53.0375 1.80962 0.904808 0.425820i \(-0.140014\pi\)
0.904808 + 0.425820i \(0.140014\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −10.2095 −0.347535 −0.173768 0.984787i \(-0.555594\pi\)
−0.173768 + 0.984787i \(0.555594\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 13.5048 0.458647
\(868\) 0 0
\(869\) −7.09539 −0.240695
\(870\) 0 0
\(871\) 10.5048 0.355941
\(872\) 0 0
\(873\) −6.00000 −0.203069
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −9.28030 −0.313374 −0.156687 0.987648i \(-0.550081\pi\)
−0.156687 + 0.987648i \(0.550081\pi\)
\(878\) 0 0
\(879\) 12.6218 0.425724
\(880\) 0 0
\(881\) −22.4923 −0.757784 −0.378892 0.925441i \(-0.623695\pi\)
−0.378892 + 0.925441i \(0.623695\pi\)
\(882\) 0 0
\(883\) 28.5972 0.962374 0.481187 0.876618i \(-0.340206\pi\)
0.481187 + 0.876618i \(0.340206\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 34.2062 1.14853 0.574265 0.818669i \(-0.305288\pi\)
0.574265 + 0.818669i \(0.305288\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −2.86464 −0.0959691
\(892\) 0 0
\(893\) −30.5048 −1.02080
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −7.52311 −0.251189
\(898\) 0 0
\(899\) 38.8188 1.29468
\(900\) 0 0
\(901\) −37.4219 −1.24670
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 25.9634 0.862099 0.431050 0.902328i \(-0.358143\pi\)
0.431050 + 0.902328i \(0.358143\pi\)
\(908\) 0 0
\(909\) −11.7293 −0.389036
\(910\) 0 0
\(911\) −51.9267 −1.72041 −0.860204 0.509949i \(-0.829664\pi\)
−0.860204 + 0.509949i \(0.829664\pi\)
\(912\) 0 0
\(913\) 23.2524 0.769542
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −11.6156 −0.383162 −0.191581 0.981477i \(-0.561362\pi\)
−0.191581 + 0.981477i \(0.561362\pi\)
\(920\) 0 0
\(921\) −20.0000 −0.659022
\(922\) 0 0
\(923\) −2.38776 −0.0785939
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −14.7755 −0.485291
\(928\) 0 0
\(929\) 5.83380 0.191401 0.0957004 0.995410i \(-0.469491\pi\)
0.0957004 + 0.995410i \(0.469491\pi\)
\(930\) 0 0
\(931\) −24.6618 −0.808258
\(932\) 0 0
\(933\) −31.8217 −1.04180
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −27.4373 −0.896338 −0.448169 0.893949i \(-0.647924\pi\)
−0.448169 + 0.893949i \(0.647924\pi\)
\(938\) 0 0
\(939\) −11.7938 −0.384877
\(940\) 0 0
\(941\) −34.9571 −1.13957 −0.569784 0.821794i \(-0.692973\pi\)
−0.569784 + 0.821794i \(0.692973\pi\)
\(942\) 0 0
\(943\) 76.1116 2.47854
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 53.3848 1.73477 0.867387 0.497634i \(-0.165798\pi\)
0.867387 + 0.497634i \(0.165798\pi\)
\(948\) 0 0
\(949\) −5.45856 −0.177192
\(950\) 0 0
\(951\) −11.4340 −0.370772
\(952\) 0 0
\(953\) −22.7110 −0.735680 −0.367840 0.929889i \(-0.619903\pi\)
−0.367840 + 0.929889i \(0.619903\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −19.4094 −0.627417
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 1.82467 0.0588603
\(962\) 0 0
\(963\) −1.72928 −0.0557253
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 28.8313 0.927153 0.463576 0.886057i \(-0.346566\pi\)
0.463576 + 0.886057i \(0.346566\pi\)
\(968\) 0 0
\(969\) −19.4586 −0.625099
\(970\) 0 0
\(971\) 19.0462 0.611223 0.305611 0.952156i \(-0.401139\pi\)
0.305611 + 0.952156i \(0.401139\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 30.1729 0.965315 0.482658 0.875809i \(-0.339672\pi\)
0.482658 + 0.875809i \(0.339672\pi\)
\(978\) 0 0
\(979\) 40.4402 1.29248
\(980\) 0 0
\(981\) 12.0279 0.384022
\(982\) 0 0
\(983\) 18.9292 0.603747 0.301874 0.953348i \(-0.402388\pi\)
0.301874 + 0.953348i \(0.402388\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −41.5510 −1.32125
\(990\) 0 0
\(991\) −25.5510 −0.811655 −0.405827 0.913950i \(-0.633017\pi\)
−0.405827 + 0.913950i \(0.633017\pi\)
\(992\) 0 0
\(993\) 2.20617 0.0700106
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 4.16952 0.132050 0.0660251 0.997818i \(-0.478968\pi\)
0.0660251 + 0.997818i \(0.478968\pi\)
\(998\) 0 0
\(999\) 3.72928 0.117989
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 7800.2.a.bp.1.2 3
5.2 odd 4 1560.2.l.c.1249.2 6
5.3 odd 4 1560.2.l.c.1249.5 yes 6
5.4 even 2 7800.2.a.bj.1.2 3
15.2 even 4 4680.2.l.e.2809.4 6
15.8 even 4 4680.2.l.e.2809.3 6
20.3 even 4 3120.2.l.m.1249.2 6
20.7 even 4 3120.2.l.m.1249.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1560.2.l.c.1249.2 6 5.2 odd 4
1560.2.l.c.1249.5 yes 6 5.3 odd 4
3120.2.l.m.1249.2 6 20.3 even 4
3120.2.l.m.1249.5 6 20.7 even 4
4680.2.l.e.2809.3 6 15.8 even 4
4680.2.l.e.2809.4 6 15.2 even 4
7800.2.a.bj.1.2 3 5.4 even 2
7800.2.a.bp.1.2 3 1.1 even 1 trivial