Properties

Label 7800.2.a.z.1.1
Level $7800$
Weight $2$
Character 7800.1
Self dual yes
Analytic conductor $62.283$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 7800.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -0.414214 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -0.414214 q^{7} +1.00000 q^{9} -2.41421 q^{11} -1.00000 q^{13} -1.82843 q^{17} -2.00000 q^{19} +0.414214 q^{21} +4.82843 q^{23} -1.00000 q^{27} +2.65685 q^{29} +1.58579 q^{31} +2.41421 q^{33} +3.65685 q^{37} +1.00000 q^{39} -5.65685 q^{41} +6.48528 q^{43} +12.0711 q^{47} -6.82843 q^{49} +1.82843 q^{51} -0.171573 q^{53} +2.00000 q^{57} -3.58579 q^{59} +3.82843 q^{61} -0.414214 q^{63} +9.24264 q^{67} -4.82843 q^{69} +3.65685 q^{71} -13.3137 q^{73} +1.00000 q^{77} -3.17157 q^{79} +1.00000 q^{81} -11.7279 q^{83} -2.65685 q^{87} +0.414214 q^{91} -1.58579 q^{93} -18.9706 q^{97} -2.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 2q^{7} + 2q^{9} - 2q^{11} - 2q^{13} + 2q^{17} - 4q^{19} - 2q^{21} + 4q^{23} - 2q^{27} - 6q^{29} + 6q^{31} + 2q^{33} - 4q^{37} + 2q^{39} - 4q^{43} + 10q^{47} - 8q^{49} - 2q^{51} - 6q^{53} + 4q^{57} - 10q^{59} + 2q^{61} + 2q^{63} + 10q^{67} - 4q^{69} - 4q^{71} - 4q^{73} + 2q^{77} - 12q^{79} + 2q^{81} + 2q^{83} + 6q^{87} - 2q^{91} - 6q^{93} - 4q^{97} - 2q^{99} + O(q^{100}) \)

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
\(7\) −0.414214 −0.156558 −0.0782790 0.996931i \(-0.524942\pi\)
−0.0782790 + 0.996931i \(0.524942\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.41421 −0.727913 −0.363956 0.931416i \(-0.618574\pi\)
−0.363956 + 0.931416i \(0.618574\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.82843 −0.443459 −0.221729 0.975108i \(-0.571170\pi\)
−0.221729 + 0.975108i \(0.571170\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 0.414214 0.0903888
\(22\) 0 0
\(23\) 4.82843 1.00680 0.503398 0.864054i \(-0.332083\pi\)
0.503398 + 0.864054i \(0.332083\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 2.65685 0.493365 0.246683 0.969096i \(-0.420659\pi\)
0.246683 + 0.969096i \(0.420659\pi\)
\(30\) 0 0
\(31\) 1.58579 0.284816 0.142408 0.989808i \(-0.454516\pi\)
0.142408 + 0.989808i \(0.454516\pi\)
\(32\) 0 0
\(33\) 2.41421 0.420261
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 3.65685 0.601183 0.300592 0.953753i \(-0.402816\pi\)
0.300592 + 0.953753i \(0.402816\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −5.65685 −0.883452 −0.441726 0.897150i \(-0.645634\pi\)
−0.441726 + 0.897150i \(0.645634\pi\)
\(42\) 0 0
\(43\) 6.48528 0.988996 0.494498 0.869179i \(-0.335352\pi\)
0.494498 + 0.869179i \(0.335352\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 12.0711 1.76075 0.880373 0.474282i \(-0.157292\pi\)
0.880373 + 0.474282i \(0.157292\pi\)
\(48\) 0 0
\(49\) −6.82843 −0.975490
\(50\) 0 0
\(51\) 1.82843 0.256031
\(52\) 0 0
\(53\) −0.171573 −0.0235673 −0.0117837 0.999931i \(-0.503751\pi\)
−0.0117837 + 0.999931i \(0.503751\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.00000 0.264906
\(58\) 0 0
\(59\) −3.58579 −0.466830 −0.233415 0.972377i \(-0.574990\pi\)
−0.233415 + 0.972377i \(0.574990\pi\)
\(60\) 0 0
\(61\) 3.82843 0.490180 0.245090 0.969500i \(-0.421183\pi\)
0.245090 + 0.969500i \(0.421183\pi\)
\(62\) 0 0
\(63\) −0.414214 −0.0521860
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 9.24264 1.12917 0.564584 0.825376i \(-0.309037\pi\)
0.564584 + 0.825376i \(0.309037\pi\)
\(68\) 0 0
\(69\) −4.82843 −0.581274
\(70\) 0 0
\(71\) 3.65685 0.433989 0.216994 0.976173i \(-0.430375\pi\)
0.216994 + 0.976173i \(0.430375\pi\)
\(72\) 0 0
\(73\) −13.3137 −1.55825 −0.779126 0.626868i \(-0.784337\pi\)
−0.779126 + 0.626868i \(0.784337\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) −3.17157 −0.356830 −0.178415 0.983955i \(-0.557097\pi\)
−0.178415 + 0.983955i \(0.557097\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −11.7279 −1.28731 −0.643653 0.765317i \(-0.722582\pi\)
−0.643653 + 0.765317i \(0.722582\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −2.65685 −0.284845
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0.414214 0.0434214
\(92\) 0 0
\(93\) −1.58579 −0.164438
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −18.9706 −1.92617 −0.963084 0.269200i \(-0.913241\pi\)
−0.963084 + 0.269200i \(0.913241\pi\)
\(98\) 0 0
\(99\) −2.41421 −0.242638
\(100\) 0 0
\(101\) −11.8284 −1.17697 −0.588486 0.808507i \(-0.700276\pi\)
−0.588486 + 0.808507i \(0.700276\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.51472 0.146433 0.0732167 0.997316i \(-0.476674\pi\)
0.0732167 + 0.997316i \(0.476674\pi\)
\(108\) 0 0
\(109\) 3.31371 0.317396 0.158698 0.987327i \(-0.449270\pi\)
0.158698 + 0.987327i \(0.449270\pi\)
\(110\) 0 0
\(111\) −3.65685 −0.347093
\(112\) 0 0
\(113\) 9.31371 0.876160 0.438080 0.898936i \(-0.355659\pi\)
0.438080 + 0.898936i \(0.355659\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) 0.757359 0.0694270
\(120\) 0 0
\(121\) −5.17157 −0.470143
\(122\) 0 0
\(123\) 5.65685 0.510061
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 16.8284 1.49328 0.746641 0.665228i \(-0.231665\pi\)
0.746641 + 0.665228i \(0.231665\pi\)
\(128\) 0 0
\(129\) −6.48528 −0.570997
\(130\) 0 0
\(131\) −6.34315 −0.554203 −0.277102 0.960841i \(-0.589374\pi\)
−0.277102 + 0.960841i \(0.589374\pi\)
\(132\) 0 0
\(133\) 0.828427 0.0718337
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 16.9706 1.44989 0.724947 0.688805i \(-0.241864\pi\)
0.724947 + 0.688805i \(0.241864\pi\)
\(138\) 0 0
\(139\) −8.82843 −0.748817 −0.374409 0.927264i \(-0.622154\pi\)
−0.374409 + 0.927264i \(0.622154\pi\)
\(140\) 0 0
\(141\) −12.0711 −1.01657
\(142\) 0 0
\(143\) 2.41421 0.201887
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 6.82843 0.563199
\(148\) 0 0
\(149\) −5.31371 −0.435316 −0.217658 0.976025i \(-0.569842\pi\)
−0.217658 + 0.976025i \(0.569842\pi\)
\(150\) 0 0
\(151\) 1.24264 0.101125 0.0505623 0.998721i \(-0.483899\pi\)
0.0505623 + 0.998721i \(0.483899\pi\)
\(152\) 0 0
\(153\) −1.82843 −0.147820
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 4.31371 0.344271 0.172136 0.985073i \(-0.444933\pi\)
0.172136 + 0.985073i \(0.444933\pi\)
\(158\) 0 0
\(159\) 0.171573 0.0136066
\(160\) 0 0
\(161\) −2.00000 −0.157622
\(162\) 0 0
\(163\) −25.3137 −1.98272 −0.991361 0.131159i \(-0.958130\pi\)
−0.991361 + 0.131159i \(0.958130\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2.68629 0.207871 0.103936 0.994584i \(-0.466856\pi\)
0.103936 + 0.994584i \(0.466856\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) 0 0
\(173\) −13.4853 −1.02527 −0.512633 0.858608i \(-0.671330\pi\)
−0.512633 + 0.858608i \(0.671330\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 3.58579 0.269524
\(178\) 0 0
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) −14.6569 −1.08944 −0.544718 0.838619i \(-0.683363\pi\)
−0.544718 + 0.838619i \(0.683363\pi\)
\(182\) 0 0
\(183\) −3.82843 −0.283005
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 4.41421 0.322799
\(188\) 0 0
\(189\) 0.414214 0.0301296
\(190\) 0 0
\(191\) −2.48528 −0.179829 −0.0899143 0.995950i \(-0.528659\pi\)
−0.0899143 + 0.995950i \(0.528659\pi\)
\(192\) 0 0
\(193\) 4.00000 0.287926 0.143963 0.989583i \(-0.454015\pi\)
0.143963 + 0.989583i \(0.454015\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −4.00000 −0.284988 −0.142494 0.989796i \(-0.545512\pi\)
−0.142494 + 0.989796i \(0.545512\pi\)
\(198\) 0 0
\(199\) 4.82843 0.342278 0.171139 0.985247i \(-0.445255\pi\)
0.171139 + 0.985247i \(0.445255\pi\)
\(200\) 0 0
\(201\) −9.24264 −0.651926
\(202\) 0 0
\(203\) −1.10051 −0.0772403
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 4.82843 0.335599
\(208\) 0 0
\(209\) 4.82843 0.333989
\(210\) 0 0
\(211\) −6.48528 −0.446465 −0.223233 0.974765i \(-0.571661\pi\)
−0.223233 + 0.974765i \(0.571661\pi\)
\(212\) 0 0
\(213\) −3.65685 −0.250564
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −0.656854 −0.0445902
\(218\) 0 0
\(219\) 13.3137 0.899657
\(220\) 0 0
\(221\) 1.82843 0.122993
\(222\) 0 0
\(223\) −12.3431 −0.826558 −0.413279 0.910604i \(-0.635617\pi\)
−0.413279 + 0.910604i \(0.635617\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 14.4142 0.956705 0.478352 0.878168i \(-0.341234\pi\)
0.478352 + 0.878168i \(0.341234\pi\)
\(228\) 0 0
\(229\) −15.6569 −1.03463 −0.517317 0.855794i \(-0.673069\pi\)
−0.517317 + 0.855794i \(0.673069\pi\)
\(230\) 0 0
\(231\) −1.00000 −0.0657952
\(232\) 0 0
\(233\) 10.0000 0.655122 0.327561 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 3.17157 0.206016
\(238\) 0 0
\(239\) 3.24264 0.209749 0.104874 0.994485i \(-0.466556\pi\)
0.104874 + 0.994485i \(0.466556\pi\)
\(240\) 0 0
\(241\) −4.00000 −0.257663 −0.128831 0.991667i \(-0.541123\pi\)
−0.128831 + 0.991667i \(0.541123\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 2.00000 0.127257
\(248\) 0 0
\(249\) 11.7279 0.743227
\(250\) 0 0
\(251\) −8.00000 −0.504956 −0.252478 0.967603i \(-0.581245\pi\)
−0.252478 + 0.967603i \(0.581245\pi\)
\(252\) 0 0
\(253\) −11.6569 −0.732860
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1.00000 0.0623783 0.0311891 0.999514i \(-0.490071\pi\)
0.0311891 + 0.999514i \(0.490071\pi\)
\(258\) 0 0
\(259\) −1.51472 −0.0941200
\(260\) 0 0
\(261\) 2.65685 0.164455
\(262\) 0 0
\(263\) 19.3137 1.19093 0.595467 0.803380i \(-0.296967\pi\)
0.595467 + 0.803380i \(0.296967\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −10.3137 −0.628838 −0.314419 0.949284i \(-0.601810\pi\)
−0.314419 + 0.949284i \(0.601810\pi\)
\(270\) 0 0
\(271\) 14.8995 0.905080 0.452540 0.891744i \(-0.350518\pi\)
0.452540 + 0.891744i \(0.350518\pi\)
\(272\) 0 0
\(273\) −0.414214 −0.0250693
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 13.3137 0.799943 0.399972 0.916528i \(-0.369020\pi\)
0.399972 + 0.916528i \(0.369020\pi\)
\(278\) 0 0
\(279\) 1.58579 0.0949386
\(280\) 0 0
\(281\) −14.0000 −0.835170 −0.417585 0.908638i \(-0.637123\pi\)
−0.417585 + 0.908638i \(0.637123\pi\)
\(282\) 0 0
\(283\) −23.4558 −1.39431 −0.697153 0.716923i \(-0.745550\pi\)
−0.697153 + 0.716923i \(0.745550\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 2.34315 0.138312
\(288\) 0 0
\(289\) −13.6569 −0.803344
\(290\) 0 0
\(291\) 18.9706 1.11207
\(292\) 0 0
\(293\) −17.3137 −1.01148 −0.505739 0.862687i \(-0.668780\pi\)
−0.505739 + 0.862687i \(0.668780\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 2.41421 0.140087
\(298\) 0 0
\(299\) −4.82843 −0.279235
\(300\) 0 0
\(301\) −2.68629 −0.154835
\(302\) 0 0
\(303\) 11.8284 0.679525
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 2.00000 0.114146 0.0570730 0.998370i \(-0.481823\pi\)
0.0570730 + 0.998370i \(0.481823\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −1.51472 −0.0858918 −0.0429459 0.999077i \(-0.513674\pi\)
−0.0429459 + 0.999077i \(0.513674\pi\)
\(312\) 0 0
\(313\) −14.1716 −0.801025 −0.400512 0.916291i \(-0.631168\pi\)
−0.400512 + 0.916291i \(0.631168\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −30.9706 −1.73948 −0.869740 0.493510i \(-0.835714\pi\)
−0.869740 + 0.493510i \(0.835714\pi\)
\(318\) 0 0
\(319\) −6.41421 −0.359127
\(320\) 0 0
\(321\) −1.51472 −0.0845433
\(322\) 0 0
\(323\) 3.65685 0.203473
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −3.31371 −0.183248
\(328\) 0 0
\(329\) −5.00000 −0.275659
\(330\) 0 0
\(331\) −14.9706 −0.822857 −0.411428 0.911442i \(-0.634970\pi\)
−0.411428 + 0.911442i \(0.634970\pi\)
\(332\) 0 0
\(333\) 3.65685 0.200394
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 9.48528 0.516696 0.258348 0.966052i \(-0.416822\pi\)
0.258348 + 0.966052i \(0.416822\pi\)
\(338\) 0 0
\(339\) −9.31371 −0.505851
\(340\) 0 0
\(341\) −3.82843 −0.207321
\(342\) 0 0
\(343\) 5.72792 0.309279
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 30.4853 1.63654 0.818268 0.574837i \(-0.194935\pi\)
0.818268 + 0.574837i \(0.194935\pi\)
\(348\) 0 0
\(349\) 7.65685 0.409862 0.204931 0.978776i \(-0.434303\pi\)
0.204931 + 0.978776i \(0.434303\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) 23.3137 1.24086 0.620432 0.784260i \(-0.286957\pi\)
0.620432 + 0.784260i \(0.286957\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −0.757359 −0.0400837
\(358\) 0 0
\(359\) −20.8995 −1.10303 −0.551517 0.834164i \(-0.685951\pi\)
−0.551517 + 0.834164i \(0.685951\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 5.17157 0.271437
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 1.65685 0.0864871 0.0432435 0.999065i \(-0.486231\pi\)
0.0432435 + 0.999065i \(0.486231\pi\)
\(368\) 0 0
\(369\) −5.65685 −0.294484
\(370\) 0 0
\(371\) 0.0710678 0.00368966
\(372\) 0 0
\(373\) −21.3431 −1.10511 −0.552553 0.833478i \(-0.686346\pi\)
−0.552553 + 0.833478i \(0.686346\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −2.65685 −0.136835
\(378\) 0 0
\(379\) −14.0711 −0.722782 −0.361391 0.932414i \(-0.617698\pi\)
−0.361391 + 0.932414i \(0.617698\pi\)
\(380\) 0 0
\(381\) −16.8284 −0.862146
\(382\) 0 0
\(383\) −7.65685 −0.391247 −0.195623 0.980679i \(-0.562673\pi\)
−0.195623 + 0.980679i \(0.562673\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 6.48528 0.329665
\(388\) 0 0
\(389\) 28.6274 1.45147 0.725734 0.687976i \(-0.241500\pi\)
0.725734 + 0.687976i \(0.241500\pi\)
\(390\) 0 0
\(391\) −8.82843 −0.446473
\(392\) 0 0
\(393\) 6.34315 0.319969
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −22.2843 −1.11842 −0.559208 0.829028i \(-0.688895\pi\)
−0.559208 + 0.829028i \(0.688895\pi\)
\(398\) 0 0
\(399\) −0.828427 −0.0414732
\(400\) 0 0
\(401\) −6.68629 −0.333897 −0.166949 0.985966i \(-0.553391\pi\)
−0.166949 + 0.985966i \(0.553391\pi\)
\(402\) 0 0
\(403\) −1.58579 −0.0789936
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −8.82843 −0.437609
\(408\) 0 0
\(409\) −3.31371 −0.163852 −0.0819262 0.996638i \(-0.526107\pi\)
−0.0819262 + 0.996638i \(0.526107\pi\)
\(410\) 0 0
\(411\) −16.9706 −0.837096
\(412\) 0 0
\(413\) 1.48528 0.0730859
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 8.82843 0.432330
\(418\) 0 0
\(419\) 13.6569 0.667181 0.333590 0.942718i \(-0.391740\pi\)
0.333590 + 0.942718i \(0.391740\pi\)
\(420\) 0 0
\(421\) 11.3137 0.551396 0.275698 0.961244i \(-0.411091\pi\)
0.275698 + 0.961244i \(0.411091\pi\)
\(422\) 0 0
\(423\) 12.0711 0.586915
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −1.58579 −0.0767416
\(428\) 0 0
\(429\) −2.41421 −0.116559
\(430\) 0 0
\(431\) −26.9706 −1.29913 −0.649563 0.760308i \(-0.725048\pi\)
−0.649563 + 0.760308i \(0.725048\pi\)
\(432\) 0 0
\(433\) −2.68629 −0.129095 −0.0645475 0.997915i \(-0.520560\pi\)
−0.0645475 + 0.997915i \(0.520560\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −9.65685 −0.461950
\(438\) 0 0
\(439\) −1.51472 −0.0722936 −0.0361468 0.999346i \(-0.511508\pi\)
−0.0361468 + 0.999346i \(0.511508\pi\)
\(440\) 0 0
\(441\) −6.82843 −0.325163
\(442\) 0 0
\(443\) −27.4558 −1.30447 −0.652233 0.758018i \(-0.726168\pi\)
−0.652233 + 0.758018i \(0.726168\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 5.31371 0.251330
\(448\) 0 0
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 13.6569 0.643076
\(452\) 0 0
\(453\) −1.24264 −0.0583844
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −18.3431 −0.858056 −0.429028 0.903291i \(-0.641144\pi\)
−0.429028 + 0.903291i \(0.641144\pi\)
\(458\) 0 0
\(459\) 1.82843 0.0853437
\(460\) 0 0
\(461\) −2.00000 −0.0931493 −0.0465746 0.998915i \(-0.514831\pi\)
−0.0465746 + 0.998915i \(0.514831\pi\)
\(462\) 0 0
\(463\) 1.72792 0.0803033 0.0401517 0.999194i \(-0.487216\pi\)
0.0401517 + 0.999194i \(0.487216\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 28.0000 1.29569 0.647843 0.761774i \(-0.275671\pi\)
0.647843 + 0.761774i \(0.275671\pi\)
\(468\) 0 0
\(469\) −3.82843 −0.176780
\(470\) 0 0
\(471\) −4.31371 −0.198765
\(472\) 0 0
\(473\) −15.6569 −0.719903
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −0.171573 −0.00785578
\(478\) 0 0
\(479\) −27.0416 −1.23556 −0.617782 0.786350i \(-0.711969\pi\)
−0.617782 + 0.786350i \(0.711969\pi\)
\(480\) 0 0
\(481\) −3.65685 −0.166738
\(482\) 0 0
\(483\) 2.00000 0.0910032
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 7.24264 0.328195 0.164098 0.986444i \(-0.447529\pi\)
0.164098 + 0.986444i \(0.447529\pi\)
\(488\) 0 0
\(489\) 25.3137 1.14473
\(490\) 0 0
\(491\) −5.51472 −0.248876 −0.124438 0.992227i \(-0.539713\pi\)
−0.124438 + 0.992227i \(0.539713\pi\)
\(492\) 0 0
\(493\) −4.85786 −0.218787
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −1.51472 −0.0679444
\(498\) 0 0
\(499\) −21.5858 −0.966313 −0.483156 0.875534i \(-0.660510\pi\)
−0.483156 + 0.875534i \(0.660510\pi\)
\(500\) 0 0
\(501\) −2.68629 −0.120015
\(502\) 0 0
\(503\) −36.2843 −1.61784 −0.808918 0.587922i \(-0.799946\pi\)
−0.808918 + 0.587922i \(0.799946\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 0 0
\(509\) −8.00000 −0.354594 −0.177297 0.984157i \(-0.556735\pi\)
−0.177297 + 0.984157i \(0.556735\pi\)
\(510\) 0 0
\(511\) 5.51472 0.243957
\(512\) 0 0
\(513\) 2.00000 0.0883022
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −29.1421 −1.28167
\(518\) 0 0
\(519\) 13.4853 0.591938
\(520\) 0 0
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 0 0
\(523\) −6.48528 −0.283582 −0.141791 0.989897i \(-0.545286\pi\)
−0.141791 + 0.989897i \(0.545286\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −2.89949 −0.126304
\(528\) 0 0
\(529\) 0.313708 0.0136395
\(530\) 0 0
\(531\) −3.58579 −0.155610
\(532\) 0 0
\(533\) 5.65685 0.245026
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 4.00000 0.172613
\(538\) 0 0
\(539\) 16.4853 0.710071
\(540\) 0 0
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) 0 0
\(543\) 14.6569 0.628986
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −22.6274 −0.967478 −0.483739 0.875212i \(-0.660722\pi\)
−0.483739 + 0.875212i \(0.660722\pi\)
\(548\) 0 0
\(549\) 3.82843 0.163393
\(550\) 0 0
\(551\) −5.31371 −0.226372
\(552\) 0 0
\(553\) 1.31371 0.0558646
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −16.0000 −0.677942 −0.338971 0.940797i \(-0.610079\pi\)
−0.338971 + 0.940797i \(0.610079\pi\)
\(558\) 0 0
\(559\) −6.48528 −0.274298
\(560\) 0 0
\(561\) −4.41421 −0.186368
\(562\) 0 0
\(563\) −6.48528 −0.273322 −0.136661 0.990618i \(-0.543637\pi\)
−0.136661 + 0.990618i \(0.543637\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −0.414214 −0.0173953
\(568\) 0 0
\(569\) 44.3137 1.85773 0.928864 0.370422i \(-0.120787\pi\)
0.928864 + 0.370422i \(0.120787\pi\)
\(570\) 0 0
\(571\) −1.51472 −0.0633890 −0.0316945 0.999498i \(-0.510090\pi\)
−0.0316945 + 0.999498i \(0.510090\pi\)
\(572\) 0 0
\(573\) 2.48528 0.103824
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 14.2843 0.594662 0.297331 0.954774i \(-0.403904\pi\)
0.297331 + 0.954774i \(0.403904\pi\)
\(578\) 0 0
\(579\) −4.00000 −0.166234
\(580\) 0 0
\(581\) 4.85786 0.201538
\(582\) 0 0
\(583\) 0.414214 0.0171550
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 13.7279 0.566612 0.283306 0.959030i \(-0.408569\pi\)
0.283306 + 0.959030i \(0.408569\pi\)
\(588\) 0 0
\(589\) −3.17157 −0.130682
\(590\) 0 0
\(591\) 4.00000 0.164538
\(592\) 0 0
\(593\) −28.6274 −1.17559 −0.587794 0.809011i \(-0.700003\pi\)
−0.587794 + 0.809011i \(0.700003\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −4.82843 −0.197614
\(598\) 0 0
\(599\) 5.65685 0.231133 0.115566 0.993300i \(-0.463132\pi\)
0.115566 + 0.993300i \(0.463132\pi\)
\(600\) 0 0
\(601\) 1.14214 0.0465887 0.0232943 0.999729i \(-0.492585\pi\)
0.0232943 + 0.999729i \(0.492585\pi\)
\(602\) 0 0
\(603\) 9.24264 0.376389
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −12.1421 −0.492834 −0.246417 0.969164i \(-0.579253\pi\)
−0.246417 + 0.969164i \(0.579253\pi\)
\(608\) 0 0
\(609\) 1.10051 0.0445947
\(610\) 0 0
\(611\) −12.0711 −0.488343
\(612\) 0 0
\(613\) −35.6569 −1.44017 −0.720083 0.693888i \(-0.755896\pi\)
−0.720083 + 0.693888i \(0.755896\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −5.02944 −0.202478 −0.101239 0.994862i \(-0.532281\pi\)
−0.101239 + 0.994862i \(0.532281\pi\)
\(618\) 0 0
\(619\) 0.627417 0.0252180 0.0126090 0.999921i \(-0.495986\pi\)
0.0126090 + 0.999921i \(0.495986\pi\)
\(620\) 0 0
\(621\) −4.82843 −0.193758
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −4.82843 −0.192829
\(628\) 0 0
\(629\) −6.68629 −0.266600
\(630\) 0 0
\(631\) 14.0000 0.557331 0.278666 0.960388i \(-0.410108\pi\)
0.278666 + 0.960388i \(0.410108\pi\)
\(632\) 0 0
\(633\) 6.48528 0.257767
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 6.82843 0.270552
\(638\) 0 0
\(639\) 3.65685 0.144663
\(640\) 0 0
\(641\) −35.4853 −1.40158 −0.700792 0.713365i \(-0.747170\pi\)
−0.700792 + 0.713365i \(0.747170\pi\)
\(642\) 0 0
\(643\) 4.62742 0.182488 0.0912438 0.995829i \(-0.470916\pi\)
0.0912438 + 0.995829i \(0.470916\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 45.9411 1.80613 0.903066 0.429502i \(-0.141311\pi\)
0.903066 + 0.429502i \(0.141311\pi\)
\(648\) 0 0
\(649\) 8.65685 0.339811
\(650\) 0 0
\(651\) 0.656854 0.0257441
\(652\) 0 0
\(653\) 34.6569 1.35623 0.678114 0.734957i \(-0.262798\pi\)
0.678114 + 0.734957i \(0.262798\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −13.3137 −0.519417
\(658\) 0 0
\(659\) −32.1421 −1.25208 −0.626040 0.779791i \(-0.715325\pi\)
−0.626040 + 0.779791i \(0.715325\pi\)
\(660\) 0 0
\(661\) −21.9411 −0.853411 −0.426705 0.904391i \(-0.640326\pi\)
−0.426705 + 0.904391i \(0.640326\pi\)
\(662\) 0 0
\(663\) −1.82843 −0.0710102
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 12.8284 0.496719
\(668\) 0 0
\(669\) 12.3431 0.477214
\(670\) 0 0
\(671\) −9.24264 −0.356808
\(672\) 0 0
\(673\) −21.1421 −0.814969 −0.407485 0.913212i \(-0.633594\pi\)
−0.407485 + 0.913212i \(0.633594\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −2.00000 −0.0768662 −0.0384331 0.999261i \(-0.512237\pi\)
−0.0384331 + 0.999261i \(0.512237\pi\)
\(678\) 0 0
\(679\) 7.85786 0.301557
\(680\) 0 0
\(681\) −14.4142 −0.552354
\(682\) 0 0
\(683\) 33.1838 1.26974 0.634871 0.772618i \(-0.281053\pi\)
0.634871 + 0.772618i \(0.281053\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 15.6569 0.597346
\(688\) 0 0
\(689\) 0.171573 0.00653641
\(690\) 0 0
\(691\) 0.757359 0.0288113 0.0144057 0.999896i \(-0.495414\pi\)
0.0144057 + 0.999896i \(0.495414\pi\)
\(692\) 0 0
\(693\) 1.00000 0.0379869
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 10.3431 0.391775
\(698\) 0 0
\(699\) −10.0000 −0.378235
\(700\) 0 0
\(701\) −9.68629 −0.365846 −0.182923 0.983127i \(-0.558556\pi\)
−0.182923 + 0.983127i \(0.558556\pi\)
\(702\) 0 0
\(703\) −7.31371 −0.275842
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 4.89949 0.184264
\(708\) 0 0
\(709\) −10.6274 −0.399121 −0.199561 0.979886i \(-0.563951\pi\)
−0.199561 + 0.979886i \(0.563951\pi\)
\(710\) 0 0
\(711\) −3.17157 −0.118943
\(712\) 0 0
\(713\) 7.65685 0.286751
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −3.24264 −0.121099
\(718\) 0 0
\(719\) 2.62742 0.0979861 0.0489931 0.998799i \(-0.484399\pi\)
0.0489931 + 0.998799i \(0.484399\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 4.00000 0.148762
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 23.3137 0.864658 0.432329 0.901716i \(-0.357692\pi\)
0.432329 + 0.901716i \(0.357692\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −11.8579 −0.438579
\(732\) 0 0
\(733\) −51.3137 −1.89532 −0.947658 0.319289i \(-0.896556\pi\)
−0.947658 + 0.319289i \(0.896556\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −22.3137 −0.821936
\(738\) 0 0
\(739\) 3.44365 0.126677 0.0633384 0.997992i \(-0.479825\pi\)
0.0633384 + 0.997992i \(0.479825\pi\)
\(740\) 0 0
\(741\) −2.00000 −0.0734718
\(742\) 0 0
\(743\) −15.5858 −0.571787 −0.285894 0.958261i \(-0.592290\pi\)
−0.285894 + 0.958261i \(0.592290\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −11.7279 −0.429102
\(748\) 0 0
\(749\) −0.627417 −0.0229253
\(750\) 0 0
\(751\) 33.5147 1.22297 0.611485 0.791256i \(-0.290573\pi\)
0.611485 + 0.791256i \(0.290573\pi\)
\(752\) 0 0
\(753\) 8.00000 0.291536
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 8.79899 0.319805 0.159902 0.987133i \(-0.448882\pi\)
0.159902 + 0.987133i \(0.448882\pi\)
\(758\) 0 0
\(759\) 11.6569 0.423117
\(760\) 0 0
\(761\) 14.0000 0.507500 0.253750 0.967270i \(-0.418336\pi\)
0.253750 + 0.967270i \(0.418336\pi\)
\(762\) 0 0
\(763\) −1.37258 −0.0496908
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 3.58579 0.129475
\(768\) 0 0
\(769\) −6.34315 −0.228740 −0.114370 0.993438i \(-0.536485\pi\)
−0.114370 + 0.993438i \(0.536485\pi\)
\(770\) 0 0
\(771\) −1.00000 −0.0360141
\(772\) 0 0
\(773\) 16.6274 0.598047 0.299023 0.954246i \(-0.403339\pi\)
0.299023 + 0.954246i \(0.403339\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 1.51472 0.0543402
\(778\) 0 0
\(779\) 11.3137 0.405356
\(780\) 0 0
\(781\) −8.82843 −0.315906
\(782\) 0 0
\(783\) −2.65685 −0.0949482
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 33.3848 1.19004 0.595019 0.803711i \(-0.297144\pi\)
0.595019 + 0.803711i \(0.297144\pi\)
\(788\) 0 0
\(789\) −19.3137 −0.687586
\(790\) 0 0
\(791\) −3.85786 −0.137170
\(792\) 0 0
\(793\) −3.82843 −0.135951
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 16.6569 0.590016 0.295008 0.955495i \(-0.404678\pi\)
0.295008 + 0.955495i \(0.404678\pi\)
\(798\) 0 0
\(799\) −22.0711 −0.780818
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 32.1421 1.13427
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 10.3137 0.363060
\(808\) 0 0
\(809\) 18.6863 0.656975 0.328488 0.944508i \(-0.393461\pi\)
0.328488 + 0.944508i \(0.393461\pi\)
\(810\) 0 0
\(811\) 53.7279 1.88664 0.943321 0.331881i \(-0.107683\pi\)
0.943321 + 0.331881i \(0.107683\pi\)
\(812\) 0 0
\(813\) −14.8995 −0.522548
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −12.9706 −0.453783
\(818\) 0 0
\(819\) 0.414214 0.0144738
\(820\) 0 0
\(821\) 10.6863 0.372954 0.186477 0.982459i \(-0.440293\pi\)
0.186477 + 0.982459i \(0.440293\pi\)
\(822\) 0 0
\(823\) −17.9411 −0.625388 −0.312694 0.949854i \(-0.601231\pi\)
−0.312694 + 0.949854i \(0.601231\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −19.3848 −0.674075 −0.337037 0.941491i \(-0.609425\pi\)
−0.337037 + 0.941491i \(0.609425\pi\)
\(828\) 0 0
\(829\) 1.82843 0.0635039 0.0317519 0.999496i \(-0.489891\pi\)
0.0317519 + 0.999496i \(0.489891\pi\)
\(830\) 0 0
\(831\) −13.3137 −0.461847
\(832\) 0 0
\(833\) 12.4853 0.432589
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −1.58579 −0.0548128
\(838\) 0 0
\(839\) −3.65685 −0.126249 −0.0631243 0.998006i \(-0.520106\pi\)
−0.0631243 + 0.998006i \(0.520106\pi\)
\(840\) 0 0
\(841\) −21.9411 −0.756591
\(842\) 0 0
\(843\) 14.0000 0.482186
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 2.14214 0.0736047
\(848\) 0 0
\(849\) 23.4558 0.805002
\(850\) 0 0
\(851\) 17.6569 0.605269
\(852\) 0 0
\(853\) −29.6569 −1.01543 −0.507716 0.861525i \(-0.669510\pi\)
−0.507716 + 0.861525i \(0.669510\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −32.6274 −1.11453 −0.557266 0.830334i \(-0.688150\pi\)
−0.557266 + 0.830334i \(0.688150\pi\)
\(858\) 0 0
\(859\) −44.8284 −1.52953 −0.764763 0.644312i \(-0.777144\pi\)
−0.764763 + 0.644312i \(0.777144\pi\)
\(860\) 0 0
\(861\) −2.34315 −0.0798542
\(862\) 0 0
\(863\) 32.4142 1.10339 0.551696 0.834045i \(-0.313981\pi\)
0.551696 + 0.834045i \(0.313981\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 13.6569 0.463811
\(868\) 0 0
\(869\) 7.65685 0.259741
\(870\) 0 0
\(871\) −9.24264 −0.313175
\(872\) 0 0
\(873\) −18.9706 −0.642056
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −25.6569 −0.866370 −0.433185 0.901305i \(-0.642610\pi\)
−0.433185 + 0.901305i \(0.642610\pi\)
\(878\) 0 0
\(879\) 17.3137 0.583977
\(880\) 0 0
\(881\) 24.5980 0.828727 0.414363 0.910111i \(-0.364004\pi\)
0.414363 + 0.910111i \(0.364004\pi\)
\(882\) 0 0
\(883\) 19.3137 0.649958 0.324979 0.945721i \(-0.394643\pi\)
0.324979 + 0.945721i \(0.394643\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 8.82843 0.296430 0.148215 0.988955i \(-0.452647\pi\)
0.148215 + 0.988955i \(0.452647\pi\)
\(888\) 0 0
\(889\) −6.97056 −0.233785
\(890\) 0 0
\(891\) −2.41421 −0.0808792
\(892\) 0 0
\(893\) −24.1421 −0.807886
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 4.82843 0.161216
\(898\) 0 0
\(899\) 4.21320 0.140518
\(900\) 0 0
\(901\) 0.313708 0.0104511
\(902\) 0 0
\(903\) 2.68629 0.0893942
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −33.9411 −1.12700 −0.563498 0.826117i \(-0.690545\pi\)
−0.563498 + 0.826117i \(0.690545\pi\)
\(908\) 0 0
\(909\) −11.8284 −0.392324
\(910\) 0 0
\(911\) −15.1716 −0.502657 −0.251328 0.967902i \(-0.580867\pi\)
−0.251328 + 0.967902i \(0.580867\pi\)
\(912\) 0 0
\(913\) 28.3137 0.937047
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 2.62742 0.0867650
\(918\) 0 0
\(919\) −21.7990 −0.719082 −0.359541 0.933129i \(-0.617067\pi\)
−0.359541 + 0.933129i \(0.617067\pi\)
\(920\) 0 0
\(921\) −2.00000 −0.0659022
\(922\) 0 0
\(923\) −3.65685 −0.120367
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 42.0000 1.37798 0.688988 0.724773i \(-0.258055\pi\)
0.688988 + 0.724773i \(0.258055\pi\)
\(930\) 0 0
\(931\) 13.6569 0.447585
\(932\) 0 0
\(933\) 1.51472 0.0495897
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −32.5980 −1.06493 −0.532465 0.846452i \(-0.678734\pi\)
−0.532465 + 0.846452i \(0.678734\pi\)
\(938\) 0 0
\(939\) 14.1716 0.462472
\(940\) 0 0
\(941\) 22.6863 0.739552 0.369776 0.929121i \(-0.379434\pi\)
0.369776 + 0.929121i \(0.379434\pi\)
\(942\) 0 0
\(943\) −27.3137 −0.889457
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 17.5858 0.571461 0.285731 0.958310i \(-0.407764\pi\)
0.285731 + 0.958310i \(0.407764\pi\)
\(948\) 0 0
\(949\) 13.3137 0.432181
\(950\) 0 0
\(951\) 30.9706 1.00429
\(952\) 0 0
\(953\) 23.3431 0.756159 0.378079 0.925773i \(-0.376585\pi\)
0.378079 + 0.925773i \(0.376585\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 6.41421 0.207342
\(958\) 0 0
\(959\) −7.02944 −0.226992
\(960\) 0 0
\(961\) −28.4853 −0.918880
\(962\) 0 0
\(963\) 1.51472 0.0488111
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −58.0122 −1.86555 −0.932773 0.360464i \(-0.882618\pi\)
−0.932773 + 0.360464i \(0.882618\pi\)
\(968\) 0 0
\(969\) −3.65685 −0.117475
\(970\) 0 0
\(971\) −23.1716 −0.743611 −0.371806 0.928311i \(-0.621261\pi\)
−0.371806 + 0.928311i \(0.621261\pi\)
\(972\) 0 0
\(973\) 3.65685 0.117233
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −0.970563 −0.0310511 −0.0155255 0.999879i \(-0.504942\pi\)
−0.0155255 + 0.999879i \(0.504942\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 3.31371 0.105799
\(982\) 0 0
\(983\) 17.2426 0.549955 0.274977 0.961451i \(-0.411330\pi\)
0.274977 + 0.961451i \(0.411330\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 5.00000 0.159152
\(988\) 0 0
\(989\) 31.3137 0.995718
\(990\) 0 0
\(991\) 44.1421 1.40222 0.701111 0.713053i \(-0.252688\pi\)
0.701111 + 0.713053i \(0.252688\pi\)
\(992\) 0 0
\(993\) 14.9706 0.475076
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −19.2010 −0.608102 −0.304051 0.952656i \(-0.598339\pi\)
−0.304051 + 0.952656i \(0.598339\pi\)
\(998\) 0 0
\(999\) −3.65685 −0.115698
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.z.1.1 2
5.4 even 2 7800.2.a.ba.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7800.2.a.z.1.1 2 1.1 even 1 trivial
7800.2.a.ba.1.2 yes 2 5.4 even 2