Properties

Label 8400.2.a.dg.1.3
Level $8400$
Weight $2$
Character 8400.1
Self dual yes
Analytic conductor $67.074$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8400 = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8400.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.0743376979\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.311108\) of defining polynomial
Character \(\chi\) \(=\) 8400.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} -2.00000 q^{11} +6.42864 q^{13} -4.42864 q^{17} +2.42864 q^{19} +1.00000 q^{21} +1.37778 q^{23} -1.00000 q^{27} +0.755569 q^{29} -5.18421 q^{31} +2.00000 q^{33} -7.61285 q^{37} -6.42864 q^{39} -8.23506 q^{41} +10.1017 q^{43} +2.75557 q^{47} +1.00000 q^{49} +4.42864 q^{51} -9.18421 q^{53} -2.42864 q^{57} -14.1017 q^{59} +6.85728 q^{61} -1.00000 q^{63} +2.75557 q^{67} -1.37778 q^{69} -2.00000 q^{71} +1.57136 q^{73} +2.00000 q^{77} +4.85728 q^{79} +1.00000 q^{81} +11.6128 q^{83} -0.755569 q^{87} +4.62222 q^{89} -6.42864 q^{91} +5.18421 q^{93} +11.9398 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} - 3 q^{7} + 3 q^{9} - 6 q^{11} + 6 q^{13} - 6 q^{19} + 3 q^{21} + 4 q^{23} - 3 q^{27} + 2 q^{29} - 2 q^{31} + 6 q^{33} + 4 q^{37} - 6 q^{39} + 2 q^{41} + 4 q^{43} + 8 q^{47} + 3 q^{49} - 14 q^{53} + 6 q^{57} - 16 q^{59} - 6 q^{61} - 3 q^{63} + 8 q^{67} - 4 q^{69} - 6 q^{71} + 18 q^{73} + 6 q^{77} - 12 q^{79} + 3 q^{81} + 8 q^{83} - 2 q^{87} + 14 q^{89} - 6 q^{91} + 2 q^{93} + 22 q^{97} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 6.42864 1.78298 0.891492 0.453037i \(-0.149659\pi\)
0.891492 + 0.453037i \(0.149659\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.42864 −1.07410 −0.537051 0.843550i \(-0.680462\pi\)
−0.537051 + 0.843550i \(0.680462\pi\)
\(18\) 0 0
\(19\) 2.42864 0.557168 0.278584 0.960412i \(-0.410135\pi\)
0.278584 + 0.960412i \(0.410135\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 1.37778 0.287288 0.143644 0.989629i \(-0.454118\pi\)
0.143644 + 0.989629i \(0.454118\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 0.755569 0.140306 0.0701528 0.997536i \(-0.477651\pi\)
0.0701528 + 0.997536i \(0.477651\pi\)
\(30\) 0 0
\(31\) −5.18421 −0.931111 −0.465556 0.885019i \(-0.654145\pi\)
−0.465556 + 0.885019i \(0.654145\pi\)
\(32\) 0 0
\(33\) 2.00000 0.348155
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −7.61285 −1.25154 −0.625772 0.780006i \(-0.715216\pi\)
−0.625772 + 0.780006i \(0.715216\pi\)
\(38\) 0 0
\(39\) −6.42864 −1.02941
\(40\) 0 0
\(41\) −8.23506 −1.28610 −0.643050 0.765824i \(-0.722331\pi\)
−0.643050 + 0.765824i \(0.722331\pi\)
\(42\) 0 0
\(43\) 10.1017 1.54050 0.770248 0.637744i \(-0.220132\pi\)
0.770248 + 0.637744i \(0.220132\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.75557 0.401941 0.200971 0.979597i \(-0.435590\pi\)
0.200971 + 0.979597i \(0.435590\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 4.42864 0.620134
\(52\) 0 0
\(53\) −9.18421 −1.26155 −0.630774 0.775967i \(-0.717263\pi\)
−0.630774 + 0.775967i \(0.717263\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −2.42864 −0.321681
\(58\) 0 0
\(59\) −14.1017 −1.83589 −0.917943 0.396712i \(-0.870151\pi\)
−0.917943 + 0.396712i \(0.870151\pi\)
\(60\) 0 0
\(61\) 6.85728 0.877985 0.438992 0.898491i \(-0.355336\pi\)
0.438992 + 0.898491i \(0.355336\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 2.75557 0.336646 0.168323 0.985732i \(-0.446165\pi\)
0.168323 + 0.985732i \(0.446165\pi\)
\(68\) 0 0
\(69\) −1.37778 −0.165866
\(70\) 0 0
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 0 0
\(73\) 1.57136 0.183914 0.0919569 0.995763i \(-0.470688\pi\)
0.0919569 + 0.995763i \(0.470688\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.00000 0.227921
\(78\) 0 0
\(79\) 4.85728 0.546487 0.273243 0.961945i \(-0.411904\pi\)
0.273243 + 0.961945i \(0.411904\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 11.6128 1.27468 0.637338 0.770585i \(-0.280036\pi\)
0.637338 + 0.770585i \(0.280036\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −0.755569 −0.0810055
\(88\) 0 0
\(89\) 4.62222 0.489954 0.244977 0.969529i \(-0.421220\pi\)
0.244977 + 0.969529i \(0.421220\pi\)
\(90\) 0 0
\(91\) −6.42864 −0.673905
\(92\) 0 0
\(93\) 5.18421 0.537577
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 11.9398 1.21230 0.606150 0.795350i \(-0.292713\pi\)
0.606150 + 0.795350i \(0.292713\pi\)
\(98\) 0 0
\(99\) −2.00000 −0.201008
\(100\) 0 0
\(101\) 1.47949 0.147215 0.0736076 0.997287i \(-0.476549\pi\)
0.0736076 + 0.997287i \(0.476549\pi\)
\(102\) 0 0
\(103\) −8.85728 −0.872734 −0.436367 0.899769i \(-0.643735\pi\)
−0.436367 + 0.899769i \(0.643735\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.76494 0.170623 0.0853114 0.996354i \(-0.472811\pi\)
0.0853114 + 0.996354i \(0.472811\pi\)
\(108\) 0 0
\(109\) 5.61285 0.537613 0.268807 0.963194i \(-0.413371\pi\)
0.268807 + 0.963194i \(0.413371\pi\)
\(110\) 0 0
\(111\) 7.61285 0.722580
\(112\) 0 0
\(113\) 11.2859 1.06169 0.530845 0.847469i \(-0.321875\pi\)
0.530845 + 0.847469i \(0.321875\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 6.42864 0.594328
\(118\) 0 0
\(119\) 4.42864 0.405973
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 8.23506 0.742531
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −12.8573 −1.14090 −0.570450 0.821333i \(-0.693231\pi\)
−0.570450 + 0.821333i \(0.693231\pi\)
\(128\) 0 0
\(129\) −10.1017 −0.889406
\(130\) 0 0
\(131\) 2.10171 0.183627 0.0918136 0.995776i \(-0.470734\pi\)
0.0918136 + 0.995776i \(0.470734\pi\)
\(132\) 0 0
\(133\) −2.42864 −0.210590
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −15.9398 −1.36183 −0.680914 0.732364i \(-0.738417\pi\)
−0.680914 + 0.732364i \(0.738417\pi\)
\(138\) 0 0
\(139\) −11.6731 −0.990097 −0.495048 0.868865i \(-0.664850\pi\)
−0.495048 + 0.868865i \(0.664850\pi\)
\(140\) 0 0
\(141\) −2.75557 −0.232061
\(142\) 0 0
\(143\) −12.8573 −1.07518
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) −21.2257 −1.73888 −0.869438 0.494041i \(-0.835519\pi\)
−0.869438 + 0.494041i \(0.835519\pi\)
\(150\) 0 0
\(151\) −16.8573 −1.37183 −0.685913 0.727684i \(-0.740597\pi\)
−0.685913 + 0.727684i \(0.740597\pi\)
\(152\) 0 0
\(153\) −4.42864 −0.358034
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 10.4286 0.832296 0.416148 0.909297i \(-0.363380\pi\)
0.416148 + 0.909297i \(0.363380\pi\)
\(158\) 0 0
\(159\) 9.18421 0.728355
\(160\) 0 0
\(161\) −1.37778 −0.108585
\(162\) 0 0
\(163\) 20.8573 1.63367 0.816834 0.576873i \(-0.195727\pi\)
0.816834 + 0.576873i \(0.195727\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 15.3461 1.18752 0.593760 0.804642i \(-0.297643\pi\)
0.593760 + 0.804642i \(0.297643\pi\)
\(168\) 0 0
\(169\) 28.3274 2.17903
\(170\) 0 0
\(171\) 2.42864 0.185723
\(172\) 0 0
\(173\) −2.06022 −0.156636 −0.0783179 0.996928i \(-0.524955\pi\)
−0.0783179 + 0.996928i \(0.524955\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 14.1017 1.05995
\(178\) 0 0
\(179\) −10.0000 −0.747435 −0.373718 0.927543i \(-0.621917\pi\)
−0.373718 + 0.927543i \(0.621917\pi\)
\(180\) 0 0
\(181\) −12.1017 −0.899513 −0.449757 0.893151i \(-0.648489\pi\)
−0.449757 + 0.893151i \(0.648489\pi\)
\(182\) 0 0
\(183\) −6.85728 −0.506905
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 8.85728 0.647708
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −0.488863 −0.0353729 −0.0176864 0.999844i \(-0.505630\pi\)
−0.0176864 + 0.999844i \(0.505630\pi\)
\(192\) 0 0
\(193\) 22.9590 1.65262 0.826312 0.563212i \(-0.190435\pi\)
0.826312 + 0.563212i \(0.190435\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1.18421 0.0843713 0.0421857 0.999110i \(-0.486568\pi\)
0.0421857 + 0.999110i \(0.486568\pi\)
\(198\) 0 0
\(199\) −8.79706 −0.623607 −0.311803 0.950147i \(-0.600933\pi\)
−0.311803 + 0.950147i \(0.600933\pi\)
\(200\) 0 0
\(201\) −2.75557 −0.194363
\(202\) 0 0
\(203\) −0.755569 −0.0530305
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 1.37778 0.0957626
\(208\) 0 0
\(209\) −4.85728 −0.335985
\(210\) 0 0
\(211\) −23.2257 −1.59892 −0.799461 0.600717i \(-0.794882\pi\)
−0.799461 + 0.600717i \(0.794882\pi\)
\(212\) 0 0
\(213\) 2.00000 0.137038
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 5.18421 0.351927
\(218\) 0 0
\(219\) −1.57136 −0.106183
\(220\) 0 0
\(221\) −28.4701 −1.91511
\(222\) 0 0
\(223\) −15.2257 −1.01959 −0.509794 0.860297i \(-0.670278\pi\)
−0.509794 + 0.860297i \(0.670278\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −14.3684 −0.953665 −0.476833 0.878994i \(-0.658215\pi\)
−0.476833 + 0.878994i \(0.658215\pi\)
\(228\) 0 0
\(229\) −5.61285 −0.370907 −0.185454 0.982653i \(-0.559375\pi\)
−0.185454 + 0.982653i \(0.559375\pi\)
\(230\) 0 0
\(231\) −2.00000 −0.131590
\(232\) 0 0
\(233\) −23.2859 −1.52551 −0.762756 0.646687i \(-0.776154\pi\)
−0.762756 + 0.646687i \(0.776154\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −4.85728 −0.315514
\(238\) 0 0
\(239\) 8.48886 0.549099 0.274549 0.961573i \(-0.411471\pi\)
0.274549 + 0.961573i \(0.411471\pi\)
\(240\) 0 0
\(241\) −7.24443 −0.466655 −0.233327 0.972398i \(-0.574961\pi\)
−0.233327 + 0.972398i \(0.574961\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 15.6128 0.993422
\(248\) 0 0
\(249\) −11.6128 −0.735934
\(250\) 0 0
\(251\) −27.6128 −1.74291 −0.871454 0.490478i \(-0.836822\pi\)
−0.871454 + 0.490478i \(0.836822\pi\)
\(252\) 0 0
\(253\) −2.75557 −0.173241
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0.428639 0.0267378 0.0133689 0.999911i \(-0.495744\pi\)
0.0133689 + 0.999911i \(0.495744\pi\)
\(258\) 0 0
\(259\) 7.61285 0.473039
\(260\) 0 0
\(261\) 0.755569 0.0467685
\(262\) 0 0
\(263\) −9.37778 −0.578259 −0.289129 0.957290i \(-0.593366\pi\)
−0.289129 + 0.957290i \(0.593366\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −4.62222 −0.282875
\(268\) 0 0
\(269\) 1.74620 0.106468 0.0532339 0.998582i \(-0.483047\pi\)
0.0532339 + 0.998582i \(0.483047\pi\)
\(270\) 0 0
\(271\) 2.69535 0.163731 0.0818653 0.996643i \(-0.473912\pi\)
0.0818653 + 0.996643i \(0.473912\pi\)
\(272\) 0 0
\(273\) 6.42864 0.389079
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 5.12399 0.307870 0.153935 0.988081i \(-0.450805\pi\)
0.153935 + 0.988081i \(0.450805\pi\)
\(278\) 0 0
\(279\) −5.18421 −0.310370
\(280\) 0 0
\(281\) 23.9813 1.43060 0.715301 0.698816i \(-0.246290\pi\)
0.715301 + 0.698816i \(0.246290\pi\)
\(282\) 0 0
\(283\) 2.36842 0.140788 0.0703939 0.997519i \(-0.477574\pi\)
0.0703939 + 0.997519i \(0.477574\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 8.23506 0.486100
\(288\) 0 0
\(289\) 2.61285 0.153697
\(290\) 0 0
\(291\) −11.9398 −0.699922
\(292\) 0 0
\(293\) −8.42864 −0.492406 −0.246203 0.969218i \(-0.579183\pi\)
−0.246203 + 0.969218i \(0.579183\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 2.00000 0.116052
\(298\) 0 0
\(299\) 8.85728 0.512230
\(300\) 0 0
\(301\) −10.1017 −0.582253
\(302\) 0 0
\(303\) −1.47949 −0.0849947
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −22.5718 −1.28824 −0.644121 0.764923i \(-0.722777\pi\)
−0.644121 + 0.764923i \(0.722777\pi\)
\(308\) 0 0
\(309\) 8.85728 0.503873
\(310\) 0 0
\(311\) 24.0830 1.36562 0.682810 0.730596i \(-0.260758\pi\)
0.682810 + 0.730596i \(0.260758\pi\)
\(312\) 0 0
\(313\) −9.65433 −0.545695 −0.272848 0.962057i \(-0.587965\pi\)
−0.272848 + 0.962057i \(0.587965\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 6.04149 0.339324 0.169662 0.985502i \(-0.445732\pi\)
0.169662 + 0.985502i \(0.445732\pi\)
\(318\) 0 0
\(319\) −1.51114 −0.0846075
\(320\) 0 0
\(321\) −1.76494 −0.0985092
\(322\) 0 0
\(323\) −10.7556 −0.598456
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −5.61285 −0.310391
\(328\) 0 0
\(329\) −2.75557 −0.151919
\(330\) 0 0
\(331\) −13.5111 −0.742639 −0.371320 0.928505i \(-0.621095\pi\)
−0.371320 + 0.928505i \(0.621095\pi\)
\(332\) 0 0
\(333\) −7.61285 −0.417181
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −10.4889 −0.571365 −0.285682 0.958324i \(-0.592220\pi\)
−0.285682 + 0.958324i \(0.592220\pi\)
\(338\) 0 0
\(339\) −11.2859 −0.612967
\(340\) 0 0
\(341\) 10.3684 0.561481
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −16.7239 −0.897787 −0.448894 0.893585i \(-0.648182\pi\)
−0.448894 + 0.893585i \(0.648182\pi\)
\(348\) 0 0
\(349\) −16.3684 −0.876181 −0.438091 0.898931i \(-0.644345\pi\)
−0.438091 + 0.898931i \(0.644345\pi\)
\(350\) 0 0
\(351\) −6.42864 −0.343135
\(352\) 0 0
\(353\) −0.549086 −0.0292249 −0.0146124 0.999893i \(-0.504651\pi\)
−0.0146124 + 0.999893i \(0.504651\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −4.42864 −0.234388
\(358\) 0 0
\(359\) 0.285442 0.0150651 0.00753253 0.999972i \(-0.497602\pi\)
0.00753253 + 0.999972i \(0.497602\pi\)
\(360\) 0 0
\(361\) −13.1017 −0.689564
\(362\) 0 0
\(363\) 7.00000 0.367405
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 1.71456 0.0894992 0.0447496 0.998998i \(-0.485751\pi\)
0.0447496 + 0.998998i \(0.485751\pi\)
\(368\) 0 0
\(369\) −8.23506 −0.428700
\(370\) 0 0
\(371\) 9.18421 0.476820
\(372\) 0 0
\(373\) −16.0000 −0.828449 −0.414224 0.910175i \(-0.635947\pi\)
−0.414224 + 0.910175i \(0.635947\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 4.85728 0.250163
\(378\) 0 0
\(379\) 4.85728 0.249502 0.124751 0.992188i \(-0.460187\pi\)
0.124751 + 0.992188i \(0.460187\pi\)
\(380\) 0 0
\(381\) 12.8573 0.658698
\(382\) 0 0
\(383\) −8.38715 −0.428563 −0.214282 0.976772i \(-0.568741\pi\)
−0.214282 + 0.976772i \(0.568741\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 10.1017 0.513499
\(388\) 0 0
\(389\) 8.95899 0.454239 0.227119 0.973867i \(-0.427069\pi\)
0.227119 + 0.973867i \(0.427069\pi\)
\(390\) 0 0
\(391\) −6.10171 −0.308577
\(392\) 0 0
\(393\) −2.10171 −0.106017
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 2.54909 0.127935 0.0639675 0.997952i \(-0.479625\pi\)
0.0639675 + 0.997952i \(0.479625\pi\)
\(398\) 0 0
\(399\) 2.42864 0.121584
\(400\) 0 0
\(401\) 0.958989 0.0478896 0.0239448 0.999713i \(-0.492377\pi\)
0.0239448 + 0.999713i \(0.492377\pi\)
\(402\) 0 0
\(403\) −33.3274 −1.66016
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 15.2257 0.754710
\(408\) 0 0
\(409\) −31.9813 −1.58137 −0.790686 0.612222i \(-0.790276\pi\)
−0.790686 + 0.612222i \(0.790276\pi\)
\(410\) 0 0
\(411\) 15.9398 0.786251
\(412\) 0 0
\(413\) 14.1017 0.693900
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 11.6731 0.571633
\(418\) 0 0
\(419\) 0.470127 0.0229672 0.0114836 0.999934i \(-0.496345\pi\)
0.0114836 + 0.999934i \(0.496345\pi\)
\(420\) 0 0
\(421\) −33.6128 −1.63819 −0.819095 0.573658i \(-0.805524\pi\)
−0.819095 + 0.573658i \(0.805524\pi\)
\(422\) 0 0
\(423\) 2.75557 0.133980
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −6.85728 −0.331847
\(428\) 0 0
\(429\) 12.8573 0.620755
\(430\) 0 0
\(431\) −11.7146 −0.564270 −0.282135 0.959375i \(-0.591043\pi\)
−0.282135 + 0.959375i \(0.591043\pi\)
\(432\) 0 0
\(433\) 0.0602231 0.00289414 0.00144707 0.999999i \(-0.499539\pi\)
0.00144707 + 0.999999i \(0.499539\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3.34614 0.160068
\(438\) 0 0
\(439\) 22.4286 1.07046 0.535230 0.844706i \(-0.320225\pi\)
0.535230 + 0.844706i \(0.320225\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −23.9496 −1.13788 −0.568940 0.822379i \(-0.692647\pi\)
−0.568940 + 0.822379i \(0.692647\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 21.2257 1.00394
\(448\) 0 0
\(449\) 29.4291 1.38885 0.694423 0.719567i \(-0.255660\pi\)
0.694423 + 0.719567i \(0.255660\pi\)
\(450\) 0 0
\(451\) 16.4701 0.775548
\(452\) 0 0
\(453\) 16.8573 0.792024
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 3.14272 0.147010 0.0735051 0.997295i \(-0.476581\pi\)
0.0735051 + 0.997295i \(0.476581\pi\)
\(458\) 0 0
\(459\) 4.42864 0.206711
\(460\) 0 0
\(461\) −3.37778 −0.157319 −0.0786596 0.996902i \(-0.525064\pi\)
−0.0786596 + 0.996902i \(0.525064\pi\)
\(462\) 0 0
\(463\) 20.8573 0.969320 0.484660 0.874703i \(-0.338943\pi\)
0.484660 + 0.874703i \(0.338943\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −14.3684 −0.664891 −0.332446 0.943122i \(-0.607874\pi\)
−0.332446 + 0.943122i \(0.607874\pi\)
\(468\) 0 0
\(469\) −2.75557 −0.127240
\(470\) 0 0
\(471\) −10.4286 −0.480526
\(472\) 0 0
\(473\) −20.2034 −0.928954
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −9.18421 −0.420516
\(478\) 0 0
\(479\) −6.36842 −0.290980 −0.145490 0.989360i \(-0.546476\pi\)
−0.145490 + 0.989360i \(0.546476\pi\)
\(480\) 0 0
\(481\) −48.9403 −2.23148
\(482\) 0 0
\(483\) 1.37778 0.0626914
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 17.3274 0.785180 0.392590 0.919714i \(-0.371579\pi\)
0.392590 + 0.919714i \(0.371579\pi\)
\(488\) 0 0
\(489\) −20.8573 −0.943199
\(490\) 0 0
\(491\) −2.00000 −0.0902587 −0.0451294 0.998981i \(-0.514370\pi\)
−0.0451294 + 0.998981i \(0.514370\pi\)
\(492\) 0 0
\(493\) −3.34614 −0.150703
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2.00000 0.0897123
\(498\) 0 0
\(499\) −23.3461 −1.04512 −0.522558 0.852603i \(-0.675022\pi\)
−0.522558 + 0.852603i \(0.675022\pi\)
\(500\) 0 0
\(501\) −15.3461 −0.685615
\(502\) 0 0
\(503\) 0.387152 0.0172623 0.00863113 0.999963i \(-0.497253\pi\)
0.00863113 + 0.999963i \(0.497253\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −28.3274 −1.25806
\(508\) 0 0
\(509\) −29.9496 −1.32749 −0.663747 0.747957i \(-0.731035\pi\)
−0.663747 + 0.747957i \(0.731035\pi\)
\(510\) 0 0
\(511\) −1.57136 −0.0695129
\(512\) 0 0
\(513\) −2.42864 −0.107227
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −5.51114 −0.242380
\(518\) 0 0
\(519\) 2.06022 0.0904338
\(520\) 0 0
\(521\) −18.5205 −0.811398 −0.405699 0.914007i \(-0.632972\pi\)
−0.405699 + 0.914007i \(0.632972\pi\)
\(522\) 0 0
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 22.9590 1.00011
\(528\) 0 0
\(529\) −21.1017 −0.917466
\(530\) 0 0
\(531\) −14.1017 −0.611962
\(532\) 0 0
\(533\) −52.9403 −2.29310
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 10.0000 0.431532
\(538\) 0 0
\(539\) −2.00000 −0.0861461
\(540\) 0 0
\(541\) 14.5906 0.627298 0.313649 0.949539i \(-0.398449\pi\)
0.313649 + 0.949539i \(0.398449\pi\)
\(542\) 0 0
\(543\) 12.1017 0.519334
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −18.7556 −0.801930 −0.400965 0.916093i \(-0.631325\pi\)
−0.400965 + 0.916093i \(0.631325\pi\)
\(548\) 0 0
\(549\) 6.85728 0.292662
\(550\) 0 0
\(551\) 1.83500 0.0781738
\(552\) 0 0
\(553\) −4.85728 −0.206553
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −31.8765 −1.35065 −0.675325 0.737520i \(-0.735997\pi\)
−0.675325 + 0.737520i \(0.735997\pi\)
\(558\) 0 0
\(559\) 64.9403 2.74668
\(560\) 0 0
\(561\) −8.85728 −0.373955
\(562\) 0 0
\(563\) −2.01874 −0.0850796 −0.0425398 0.999095i \(-0.513545\pi\)
−0.0425398 + 0.999095i \(0.513545\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) −28.9590 −1.21402 −0.607012 0.794693i \(-0.707632\pi\)
−0.607012 + 0.794693i \(0.707632\pi\)
\(570\) 0 0
\(571\) −8.97773 −0.375706 −0.187853 0.982197i \(-0.560153\pi\)
−0.187853 + 0.982197i \(0.560153\pi\)
\(572\) 0 0
\(573\) 0.488863 0.0204225
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −28.6766 −1.19382 −0.596911 0.802307i \(-0.703606\pi\)
−0.596911 + 0.802307i \(0.703606\pi\)
\(578\) 0 0
\(579\) −22.9590 −0.954143
\(580\) 0 0
\(581\) −11.6128 −0.481782
\(582\) 0 0
\(583\) 18.3684 0.760742
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 45.2070 1.86589 0.932945 0.360018i \(-0.117229\pi\)
0.932945 + 0.360018i \(0.117229\pi\)
\(588\) 0 0
\(589\) −12.5906 −0.518786
\(590\) 0 0
\(591\) −1.18421 −0.0487118
\(592\) 0 0
\(593\) 18.2636 0.749998 0.374999 0.927025i \(-0.377643\pi\)
0.374999 + 0.927025i \(0.377643\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 8.79706 0.360040
\(598\) 0 0
\(599\) 22.7368 0.929002 0.464501 0.885573i \(-0.346234\pi\)
0.464501 + 0.885573i \(0.346234\pi\)
\(600\) 0 0
\(601\) 0.488863 0.0199411 0.00997056 0.999950i \(-0.496826\pi\)
0.00997056 + 0.999950i \(0.496826\pi\)
\(602\) 0 0
\(603\) 2.75557 0.112215
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 20.2034 0.820032 0.410016 0.912078i \(-0.365523\pi\)
0.410016 + 0.912078i \(0.365523\pi\)
\(608\) 0 0
\(609\) 0.755569 0.0306172
\(610\) 0 0
\(611\) 17.7146 0.716654
\(612\) 0 0
\(613\) 10.3684 0.418776 0.209388 0.977833i \(-0.432853\pi\)
0.209388 + 0.977833i \(0.432853\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −39.2859 −1.58159 −0.790796 0.612080i \(-0.790333\pi\)
−0.790796 + 0.612080i \(0.790333\pi\)
\(618\) 0 0
\(619\) −42.8988 −1.72425 −0.862123 0.506698i \(-0.830866\pi\)
−0.862123 + 0.506698i \(0.830866\pi\)
\(620\) 0 0
\(621\) −1.37778 −0.0552886
\(622\) 0 0
\(623\) −4.62222 −0.185185
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 4.85728 0.193981
\(628\) 0 0
\(629\) 33.7146 1.34429
\(630\) 0 0
\(631\) −15.3461 −0.610920 −0.305460 0.952205i \(-0.598810\pi\)
−0.305460 + 0.952205i \(0.598810\pi\)
\(632\) 0 0
\(633\) 23.2257 0.923139
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 6.42864 0.254712
\(638\) 0 0
\(639\) −2.00000 −0.0791188
\(640\) 0 0
\(641\) 30.6735 1.21153 0.605766 0.795643i \(-0.292867\pi\)
0.605766 + 0.795643i \(0.292867\pi\)
\(642\) 0 0
\(643\) 49.0607 1.93477 0.967383 0.253320i \(-0.0815225\pi\)
0.967383 + 0.253320i \(0.0815225\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 15.3461 0.603319 0.301660 0.953416i \(-0.402459\pi\)
0.301660 + 0.953416i \(0.402459\pi\)
\(648\) 0 0
\(649\) 28.2034 1.10708
\(650\) 0 0
\(651\) −5.18421 −0.203185
\(652\) 0 0
\(653\) −19.4697 −0.761906 −0.380953 0.924594i \(-0.624404\pi\)
−0.380953 + 0.924594i \(0.624404\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 1.57136 0.0613046
\(658\) 0 0
\(659\) −30.9403 −1.20526 −0.602631 0.798020i \(-0.705881\pi\)
−0.602631 + 0.798020i \(0.705881\pi\)
\(660\) 0 0
\(661\) 47.7975 1.85911 0.929554 0.368685i \(-0.120192\pi\)
0.929554 + 0.368685i \(0.120192\pi\)
\(662\) 0 0
\(663\) 28.4701 1.10569
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1.04101 0.0403081
\(668\) 0 0
\(669\) 15.2257 0.588659
\(670\) 0 0
\(671\) −13.7146 −0.529445
\(672\) 0 0
\(673\) 27.8163 1.07224 0.536119 0.844142i \(-0.319890\pi\)
0.536119 + 0.844142i \(0.319890\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −19.0005 −0.730248 −0.365124 0.930959i \(-0.618973\pi\)
−0.365124 + 0.930959i \(0.618973\pi\)
\(678\) 0 0
\(679\) −11.9398 −0.458207
\(680\) 0 0
\(681\) 14.3684 0.550599
\(682\) 0 0
\(683\) −4.52051 −0.172972 −0.0864862 0.996253i \(-0.527564\pi\)
−0.0864862 + 0.996253i \(0.527564\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 5.61285 0.214143
\(688\) 0 0
\(689\) −59.0420 −2.24932
\(690\) 0 0
\(691\) 1.18421 0.0450494 0.0225247 0.999746i \(-0.492830\pi\)
0.0225247 + 0.999746i \(0.492830\pi\)
\(692\) 0 0
\(693\) 2.00000 0.0759737
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 36.4701 1.38140
\(698\) 0 0
\(699\) 23.2859 0.880754
\(700\) 0 0
\(701\) −26.6735 −1.00745 −0.503723 0.863865i \(-0.668037\pi\)
−0.503723 + 0.863865i \(0.668037\pi\)
\(702\) 0 0
\(703\) −18.4889 −0.697321
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −1.47949 −0.0556421
\(708\) 0 0
\(709\) −18.2034 −0.683644 −0.341822 0.939765i \(-0.611044\pi\)
−0.341822 + 0.939765i \(0.611044\pi\)
\(710\) 0 0
\(711\) 4.85728 0.182162
\(712\) 0 0
\(713\) −7.14272 −0.267497
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −8.48886 −0.317022
\(718\) 0 0
\(719\) −4.85728 −0.181146 −0.0905730 0.995890i \(-0.528870\pi\)
−0.0905730 + 0.995890i \(0.528870\pi\)
\(720\) 0 0
\(721\) 8.85728 0.329862
\(722\) 0 0
\(723\) 7.24443 0.269423
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −21.0607 −0.781098 −0.390549 0.920582i \(-0.627715\pi\)
−0.390549 + 0.920582i \(0.627715\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −44.7368 −1.65465
\(732\) 0 0
\(733\) 9.45091 0.349077 0.174539 0.984650i \(-0.444157\pi\)
0.174539 + 0.984650i \(0.444157\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −5.51114 −0.203005
\(738\) 0 0
\(739\) 8.20342 0.301768 0.150884 0.988551i \(-0.451788\pi\)
0.150884 + 0.988551i \(0.451788\pi\)
\(740\) 0 0
\(741\) −15.6128 −0.573552
\(742\) 0 0
\(743\) −8.33677 −0.305847 −0.152923 0.988238i \(-0.548869\pi\)
−0.152923 + 0.988238i \(0.548869\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 11.6128 0.424892
\(748\) 0 0
\(749\) −1.76494 −0.0644894
\(750\) 0 0
\(751\) 25.9180 0.945760 0.472880 0.881127i \(-0.343214\pi\)
0.472880 + 0.881127i \(0.343214\pi\)
\(752\) 0 0
\(753\) 27.6128 1.00627
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 8.94025 0.324939 0.162470 0.986714i \(-0.448054\pi\)
0.162470 + 0.986714i \(0.448054\pi\)
\(758\) 0 0
\(759\) 2.75557 0.100021
\(760\) 0 0
\(761\) −0.825636 −0.0299293 −0.0149646 0.999888i \(-0.504764\pi\)
−0.0149646 + 0.999888i \(0.504764\pi\)
\(762\) 0 0
\(763\) −5.61285 −0.203199
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −90.6548 −3.27336
\(768\) 0 0
\(769\) 21.2257 0.765418 0.382709 0.923869i \(-0.374991\pi\)
0.382709 + 0.923869i \(0.374991\pi\)
\(770\) 0 0
\(771\) −0.428639 −0.0154371
\(772\) 0 0
\(773\) 29.4893 1.06066 0.530329 0.847792i \(-0.322068\pi\)
0.530329 + 0.847792i \(0.322068\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −7.61285 −0.273109
\(778\) 0 0
\(779\) −20.0000 −0.716574
\(780\) 0 0
\(781\) 4.00000 0.143131
\(782\) 0 0
\(783\) −0.755569 −0.0270018
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 34.4514 1.22806 0.614030 0.789283i \(-0.289547\pi\)
0.614030 + 0.789283i \(0.289547\pi\)
\(788\) 0 0
\(789\) 9.37778 0.333858
\(790\) 0 0
\(791\) −11.2859 −0.401281
\(792\) 0 0
\(793\) 44.0830 1.56543
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 18.9175 0.670092 0.335046 0.942202i \(-0.391248\pi\)
0.335046 + 0.942202i \(0.391248\pi\)
\(798\) 0 0
\(799\) −12.2034 −0.431726
\(800\) 0 0
\(801\) 4.62222 0.163318
\(802\) 0 0
\(803\) −3.14272 −0.110904
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −1.74620 −0.0614692
\(808\) 0 0
\(809\) 21.2257 0.746256 0.373128 0.927780i \(-0.378285\pi\)
0.373128 + 0.927780i \(0.378285\pi\)
\(810\) 0 0
\(811\) 21.5081 0.755251 0.377625 0.925958i \(-0.376741\pi\)
0.377625 + 0.925958i \(0.376741\pi\)
\(812\) 0 0
\(813\) −2.69535 −0.0945299
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 24.5334 0.858315
\(818\) 0 0
\(819\) −6.42864 −0.224635
\(820\) 0 0
\(821\) −46.2034 −1.61251 −0.806255 0.591568i \(-0.798509\pi\)
−0.806255 + 0.591568i \(0.798509\pi\)
\(822\) 0 0
\(823\) 17.8350 0.621689 0.310845 0.950461i \(-0.399388\pi\)
0.310845 + 0.950461i \(0.399388\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −35.2128 −1.22447 −0.612234 0.790676i \(-0.709729\pi\)
−0.612234 + 0.790676i \(0.709729\pi\)
\(828\) 0 0
\(829\) 14.3872 0.499686 0.249843 0.968286i \(-0.419621\pi\)
0.249843 + 0.968286i \(0.419621\pi\)
\(830\) 0 0
\(831\) −5.12399 −0.177749
\(832\) 0 0
\(833\) −4.42864 −0.153443
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 5.18421 0.179192
\(838\) 0 0
\(839\) −1.51114 −0.0521703 −0.0260851 0.999660i \(-0.508304\pi\)
−0.0260851 + 0.999660i \(0.508304\pi\)
\(840\) 0 0
\(841\) −28.4291 −0.980314
\(842\) 0 0
\(843\) −23.9813 −0.825959
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 7.00000 0.240523
\(848\) 0 0
\(849\) −2.36842 −0.0812838
\(850\) 0 0
\(851\) −10.4889 −0.359554
\(852\) 0 0
\(853\) −15.4064 −0.527504 −0.263752 0.964591i \(-0.584960\pi\)
−0.263752 + 0.964591i \(0.584960\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 19.8578 0.678328 0.339164 0.940727i \(-0.389856\pi\)
0.339164 + 0.940727i \(0.389856\pi\)
\(858\) 0 0
\(859\) −2.42864 −0.0828641 −0.0414321 0.999141i \(-0.513192\pi\)
−0.0414321 + 0.999141i \(0.513192\pi\)
\(860\) 0 0
\(861\) −8.23506 −0.280650
\(862\) 0 0
\(863\) 39.2958 1.33764 0.668822 0.743423i \(-0.266799\pi\)
0.668822 + 0.743423i \(0.266799\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −2.61285 −0.0887370
\(868\) 0 0
\(869\) −9.71456 −0.329544
\(870\) 0 0
\(871\) 17.7146 0.600235
\(872\) 0 0
\(873\) 11.9398 0.404100
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −56.2864 −1.90066 −0.950328 0.311249i \(-0.899253\pi\)
−0.950328 + 0.311249i \(0.899253\pi\)
\(878\) 0 0
\(879\) 8.42864 0.284291
\(880\) 0 0
\(881\) −2.33677 −0.0787279 −0.0393640 0.999225i \(-0.512533\pi\)
−0.0393640 + 0.999225i \(0.512533\pi\)
\(882\) 0 0
\(883\) −33.7146 −1.13459 −0.567293 0.823516i \(-0.692009\pi\)
−0.567293 + 0.823516i \(0.692009\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −47.8992 −1.60830 −0.804150 0.594427i \(-0.797379\pi\)
−0.804150 + 0.594427i \(0.797379\pi\)
\(888\) 0 0
\(889\) 12.8573 0.431219
\(890\) 0 0
\(891\) −2.00000 −0.0670025
\(892\) 0 0
\(893\) 6.69228 0.223949
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −8.85728 −0.295736
\(898\) 0 0
\(899\) −3.91703 −0.130640
\(900\) 0 0
\(901\) 40.6735 1.35503
\(902\) 0 0
\(903\) 10.1017 0.336164
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −23.7591 −0.788908 −0.394454 0.918916i \(-0.629066\pi\)
−0.394454 + 0.918916i \(0.629066\pi\)
\(908\) 0 0
\(909\) 1.47949 0.0490717
\(910\) 0 0
\(911\) −22.9403 −0.760045 −0.380022 0.924977i \(-0.624084\pi\)
−0.380022 + 0.924977i \(0.624084\pi\)
\(912\) 0 0
\(913\) −23.2257 −0.768658
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −2.10171 −0.0694046
\(918\) 0 0
\(919\) −16.9777 −0.560043 −0.280022 0.959994i \(-0.590342\pi\)
−0.280022 + 0.959994i \(0.590342\pi\)
\(920\) 0 0
\(921\) 22.5718 0.743767
\(922\) 0 0
\(923\) −12.8573 −0.423202
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −8.85728 −0.290911
\(928\) 0 0
\(929\) 39.3403 1.29071 0.645357 0.763881i \(-0.276709\pi\)
0.645357 + 0.763881i \(0.276709\pi\)
\(930\) 0 0
\(931\) 2.42864 0.0795954
\(932\) 0 0
\(933\) −24.0830 −0.788441
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −17.7748 −0.580677 −0.290338 0.956924i \(-0.593768\pi\)
−0.290338 + 0.956924i \(0.593768\pi\)
\(938\) 0 0
\(939\) 9.65433 0.315057
\(940\) 0 0
\(941\) 35.5812 1.15991 0.579957 0.814647i \(-0.303069\pi\)
0.579957 + 0.814647i \(0.303069\pi\)
\(942\) 0 0
\(943\) −11.3461 −0.369481
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −30.5018 −0.991174 −0.495587 0.868558i \(-0.665047\pi\)
−0.495587 + 0.868558i \(0.665047\pi\)
\(948\) 0 0
\(949\) 10.1017 0.327915
\(950\) 0 0
\(951\) −6.04149 −0.195909
\(952\) 0 0
\(953\) −51.1655 −1.65741 −0.828706 0.559684i \(-0.810923\pi\)
−0.828706 + 0.559684i \(0.810923\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 1.51114 0.0488481
\(958\) 0 0
\(959\) 15.9398 0.514722
\(960\) 0 0
\(961\) −4.12399 −0.133032
\(962\) 0 0
\(963\) 1.76494 0.0568743
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −47.8992 −1.54034 −0.770168 0.637841i \(-0.779828\pi\)
−0.770168 + 0.637841i \(0.779828\pi\)
\(968\) 0 0
\(969\) 10.7556 0.345519
\(970\) 0 0
\(971\) −40.6735 −1.30528 −0.652638 0.757670i \(-0.726338\pi\)
−0.652638 + 0.757670i \(0.726338\pi\)
\(972\) 0 0
\(973\) 11.6731 0.374221
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −27.4893 −0.879462 −0.439731 0.898130i \(-0.644926\pi\)
−0.439731 + 0.898130i \(0.644926\pi\)
\(978\) 0 0
\(979\) −9.24443 −0.295453
\(980\) 0 0
\(981\) 5.61285 0.179204
\(982\) 0 0
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 2.75557 0.0877107
\(988\) 0 0
\(989\) 13.9180 0.442566
\(990\) 0 0
\(991\) 34.6923 1.10204 0.551018 0.834493i \(-0.314239\pi\)
0.551018 + 0.834493i \(0.314239\pi\)
\(992\) 0 0
\(993\) 13.5111 0.428763
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −28.6766 −0.908197 −0.454099 0.890951i \(-0.650039\pi\)
−0.454099 + 0.890951i \(0.650039\pi\)
\(998\) 0 0
\(999\) 7.61285 0.240860
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 8400.2.a.dg.1.3 3
4.3 odd 2 525.2.a.j.1.3 3
5.2 odd 4 1680.2.t.k.1009.5 6
5.3 odd 4 1680.2.t.k.1009.2 6
5.4 even 2 8400.2.a.dj.1.1 3
12.11 even 2 1575.2.a.x.1.1 3
15.2 even 4 5040.2.t.v.1009.3 6
15.8 even 4 5040.2.t.v.1009.4 6
20.3 even 4 105.2.d.b.64.2 6
20.7 even 4 105.2.d.b.64.5 yes 6
20.19 odd 2 525.2.a.k.1.1 3
28.27 even 2 3675.2.a.bi.1.3 3
60.23 odd 4 315.2.d.e.64.5 6
60.47 odd 4 315.2.d.e.64.2 6
60.59 even 2 1575.2.a.w.1.3 3
140.3 odd 12 735.2.q.f.79.5 12
140.23 even 12 735.2.q.e.214.2 12
140.27 odd 4 735.2.d.b.589.5 6
140.47 odd 12 735.2.q.f.214.5 12
140.67 even 12 735.2.q.e.79.2 12
140.83 odd 4 735.2.d.b.589.2 6
140.87 odd 12 735.2.q.f.79.2 12
140.103 odd 12 735.2.q.f.214.2 12
140.107 even 12 735.2.q.e.214.5 12
140.123 even 12 735.2.q.e.79.5 12
140.139 even 2 3675.2.a.bj.1.1 3
420.83 even 4 2205.2.d.l.1324.5 6
420.167 even 4 2205.2.d.l.1324.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.2.d.b.64.2 6 20.3 even 4
105.2.d.b.64.5 yes 6 20.7 even 4
315.2.d.e.64.2 6 60.47 odd 4
315.2.d.e.64.5 6 60.23 odd 4
525.2.a.j.1.3 3 4.3 odd 2
525.2.a.k.1.1 3 20.19 odd 2
735.2.d.b.589.2 6 140.83 odd 4
735.2.d.b.589.5 6 140.27 odd 4
735.2.q.e.79.2 12 140.67 even 12
735.2.q.e.79.5 12 140.123 even 12
735.2.q.e.214.2 12 140.23 even 12
735.2.q.e.214.5 12 140.107 even 12
735.2.q.f.79.2 12 140.87 odd 12
735.2.q.f.79.5 12 140.3 odd 12
735.2.q.f.214.2 12 140.103 odd 12
735.2.q.f.214.5 12 140.47 odd 12
1575.2.a.w.1.3 3 60.59 even 2
1575.2.a.x.1.1 3 12.11 even 2
1680.2.t.k.1009.2 6 5.3 odd 4
1680.2.t.k.1009.5 6 5.2 odd 4
2205.2.d.l.1324.2 6 420.167 even 4
2205.2.d.l.1324.5 6 420.83 even 4
3675.2.a.bi.1.3 3 28.27 even 2
3675.2.a.bj.1.1 3 140.139 even 2
5040.2.t.v.1009.3 6 15.2 even 4
5040.2.t.v.1009.4 6 15.8 even 4
8400.2.a.dg.1.3 3 1.1 even 1 trivial
8400.2.a.dj.1.1 3 5.4 even 2