Properties

Label 8400.2.a.dj.1.2
Level $8400$
Weight $2$
Character 8400.1
Self dual yes
Analytic conductor $67.074$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(x^{3} - x^{2} - 3 x + 1\)
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 1.2
Root \(2.17009\) of defining polynomial
Character \(\chi\) \(=\) 8400.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} -2.00000 q^{11} -0.921622 q^{13} -1.07838 q^{17} -3.07838 q^{19} +1.00000 q^{21} +2.34017 q^{23} +1.00000 q^{27} -6.68035 q^{29} +7.75872 q^{31} -2.00000 q^{33} -10.8371 q^{37} -0.921622 q^{39} +6.49693 q^{41} -6.52359 q^{43} +4.68035 q^{47} +1.00000 q^{49} -1.07838 q^{51} -3.75872 q^{53} -3.07838 q^{57} -10.5236 q^{59} -4.15676 q^{61} +1.00000 q^{63} +4.68035 q^{67} +2.34017 q^{69} -2.00000 q^{71} -7.07838 q^{73} -2.00000 q^{77} -6.15676 q^{79} +1.00000 q^{81} +6.83710 q^{83} -6.68035 q^{87} +8.34017 q^{89} -0.921622 q^{91} +7.75872 q^{93} +8.43907 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + 3 q^{7} + 3 q^{9} + O(q^{10}) \) \( 3 q + 3 q^{3} + 3 q^{7} + 3 q^{9} - 6 q^{11} - 6 q^{13} - 6 q^{19} + 3 q^{21} - 4 q^{23} + 3 q^{27} + 2 q^{29} - 2 q^{31} - 6 q^{33} - 4 q^{37} - 6 q^{39} + 2 q^{41} - 4 q^{43} - 8 q^{47} + 3 q^{49} + 14 q^{53} - 6 q^{57} - 16 q^{59} - 6 q^{61} + 3 q^{63} - 8 q^{67} - 4 q^{69} - 6 q^{71} - 18 q^{73} - 6 q^{77} - 12 q^{79} + 3 q^{81} - 8 q^{83} + 2 q^{87} + 14 q^{89} - 6 q^{91} - 2 q^{93} - 22 q^{97} - 6 q^{99} + O(q^{100}) \)

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
\(13\) −0.921622 −0.255612 −0.127806 0.991799i \(-0.540793\pi\)
−0.127806 + 0.991799i \(0.540793\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.07838 −0.261545 −0.130773 0.991412i \(-0.541746\pi\)
−0.130773 + 0.991412i \(0.541746\pi\)
\(18\) 0 0
\(19\) −3.07838 −0.706228 −0.353114 0.935580i \(-0.614877\pi\)
−0.353114 + 0.935580i \(0.614877\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 2.34017 0.487960 0.243980 0.969780i \(-0.421547\pi\)
0.243980 + 0.969780i \(0.421547\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −6.68035 −1.24051 −0.620255 0.784401i \(-0.712971\pi\)
−0.620255 + 0.784401i \(0.712971\pi\)
\(30\) 0 0
\(31\) 7.75872 1.39351 0.696754 0.717310i \(-0.254627\pi\)
0.696754 + 0.717310i \(0.254627\pi\)
\(32\) 0 0
\(33\) −2.00000 −0.348155
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −10.8371 −1.78161 −0.890804 0.454387i \(-0.849858\pi\)
−0.890804 + 0.454387i \(0.849858\pi\)
\(38\) 0 0
\(39\) −0.921622 −0.147578
\(40\) 0 0
\(41\) 6.49693 1.01465 0.507325 0.861755i \(-0.330634\pi\)
0.507325 + 0.861755i \(0.330634\pi\)
\(42\) 0 0
\(43\) −6.52359 −0.994838 −0.497419 0.867510i \(-0.665719\pi\)
−0.497419 + 0.867510i \(0.665719\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.68035 0.682699 0.341349 0.939937i \(-0.389116\pi\)
0.341349 + 0.939937i \(0.389116\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −1.07838 −0.151003
\(52\) 0 0
\(53\) −3.75872 −0.516300 −0.258150 0.966105i \(-0.583113\pi\)
−0.258150 + 0.966105i \(0.583113\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −3.07838 −0.407741
\(58\) 0 0
\(59\) −10.5236 −1.37005 −0.685027 0.728517i \(-0.740210\pi\)
−0.685027 + 0.728517i \(0.740210\pi\)
\(60\) 0 0
\(61\) −4.15676 −0.532218 −0.266109 0.963943i \(-0.585738\pi\)
−0.266109 + 0.963943i \(0.585738\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.68035 0.571795 0.285898 0.958260i \(-0.407708\pi\)
0.285898 + 0.958260i \(0.407708\pi\)
\(68\) 0 0
\(69\) 2.34017 0.281724
\(70\) 0 0
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 0 0
\(73\) −7.07838 −0.828461 −0.414231 0.910172i \(-0.635949\pi\)
−0.414231 + 0.910172i \(0.635949\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.00000 −0.227921
\(78\) 0 0
\(79\) −6.15676 −0.692689 −0.346345 0.938107i \(-0.612577\pi\)
−0.346345 + 0.938107i \(0.612577\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 6.83710 0.750469 0.375235 0.926930i \(-0.377562\pi\)
0.375235 + 0.926930i \(0.377562\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −6.68035 −0.716208
\(88\) 0 0
\(89\) 8.34017 0.884057 0.442028 0.897001i \(-0.354259\pi\)
0.442028 + 0.897001i \(0.354259\pi\)
\(90\) 0 0
\(91\) −0.921622 −0.0966123
\(92\) 0 0
\(93\) 7.75872 0.804542
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 8.43907 0.856858 0.428429 0.903576i \(-0.359067\pi\)
0.428429 + 0.903576i \(0.359067\pi\)
\(98\) 0 0
\(99\) −2.00000 −0.201008
\(100\) 0 0
\(101\) −5.81658 −0.578772 −0.289386 0.957213i \(-0.593451\pi\)
−0.289386 + 0.957213i \(0.593451\pi\)
\(102\) 0 0
\(103\) −2.15676 −0.212511 −0.106256 0.994339i \(-0.533886\pi\)
−0.106256 + 0.994339i \(0.533886\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −16.4969 −1.59482 −0.797409 0.603439i \(-0.793797\pi\)
−0.797409 + 0.603439i \(0.793797\pi\)
\(108\) 0 0
\(109\) −12.8371 −1.22957 −0.614786 0.788694i \(-0.710757\pi\)
−0.614786 + 0.788694i \(0.710757\pi\)
\(110\) 0 0
\(111\) −10.8371 −1.02861
\(112\) 0 0
\(113\) 5.23513 0.492480 0.246240 0.969209i \(-0.420805\pi\)
0.246240 + 0.969209i \(0.420805\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −0.921622 −0.0852040
\(118\) 0 0
\(119\) −1.07838 −0.0988547
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 6.49693 0.585808
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 1.84324 0.163562 0.0817808 0.996650i \(-0.473939\pi\)
0.0817808 + 0.996650i \(0.473939\pi\)
\(128\) 0 0
\(129\) −6.52359 −0.574370
\(130\) 0 0
\(131\) −1.47641 −0.128995 −0.0644973 0.997918i \(-0.520544\pi\)
−0.0644973 + 0.997918i \(0.520544\pi\)
\(132\) 0 0
\(133\) −3.07838 −0.266929
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −4.43907 −0.379255 −0.189628 0.981856i \(-0.560728\pi\)
−0.189628 + 0.981856i \(0.560728\pi\)
\(138\) 0 0
\(139\) −13.6020 −1.15370 −0.576852 0.816849i \(-0.695719\pi\)
−0.576852 + 0.816849i \(0.695719\pi\)
\(140\) 0 0
\(141\) 4.68035 0.394156
\(142\) 0 0
\(143\) 1.84324 0.154140
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) 15.6742 1.28408 0.642040 0.766671i \(-0.278088\pi\)
0.642040 + 0.766671i \(0.278088\pi\)
\(150\) 0 0
\(151\) −5.84324 −0.475516 −0.237758 0.971324i \(-0.576413\pi\)
−0.237758 + 0.971324i \(0.576413\pi\)
\(152\) 0 0
\(153\) −1.07838 −0.0871817
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −4.92162 −0.392788 −0.196394 0.980525i \(-0.562923\pi\)
−0.196394 + 0.980525i \(0.562923\pi\)
\(158\) 0 0
\(159\) −3.75872 −0.298086
\(160\) 0 0
\(161\) 2.34017 0.184431
\(162\) 0 0
\(163\) −9.84324 −0.770982 −0.385491 0.922712i \(-0.625968\pi\)
−0.385491 + 0.922712i \(0.625968\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −19.2039 −1.48605 −0.743023 0.669266i \(-0.766609\pi\)
−0.743023 + 0.669266i \(0.766609\pi\)
\(168\) 0 0
\(169\) −12.1506 −0.934662
\(170\) 0 0
\(171\) −3.07838 −0.235409
\(172\) 0 0
\(173\) 22.4391 1.70601 0.853005 0.521902i \(-0.174777\pi\)
0.853005 + 0.521902i \(0.174777\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −10.5236 −0.791001
\(178\) 0 0
\(179\) −10.0000 −0.747435 −0.373718 0.927543i \(-0.621917\pi\)
−0.373718 + 0.927543i \(0.621917\pi\)
\(180\) 0 0
\(181\) −8.52359 −0.633553 −0.316777 0.948500i \(-0.602601\pi\)
−0.316777 + 0.948500i \(0.602601\pi\)
\(182\) 0 0
\(183\) −4.15676 −0.307276
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 2.15676 0.157718
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −15.3607 −1.11146 −0.555730 0.831363i \(-0.687561\pi\)
−0.555730 + 0.831363i \(0.687561\pi\)
\(192\) 0 0
\(193\) −8.36683 −0.602258 −0.301129 0.953583i \(-0.597363\pi\)
−0.301129 + 0.953583i \(0.597363\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 11.7587 0.837774 0.418887 0.908038i \(-0.362420\pi\)
0.418887 + 0.908038i \(0.362420\pi\)
\(198\) 0 0
\(199\) 22.5958 1.60178 0.800888 0.598814i \(-0.204361\pi\)
0.800888 + 0.598814i \(0.204361\pi\)
\(200\) 0 0
\(201\) 4.68035 0.330126
\(202\) 0 0
\(203\) −6.68035 −0.468868
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 2.34017 0.162653
\(208\) 0 0
\(209\) 6.15676 0.425872
\(210\) 0 0
\(211\) 13.6742 0.941371 0.470685 0.882301i \(-0.344007\pi\)
0.470685 + 0.882301i \(0.344007\pi\)
\(212\) 0 0
\(213\) −2.00000 −0.137038
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 7.75872 0.526696
\(218\) 0 0
\(219\) −7.07838 −0.478312
\(220\) 0 0
\(221\) 0.993857 0.0668541
\(222\) 0 0
\(223\) −21.6742 −1.45141 −0.725706 0.688005i \(-0.758487\pi\)
−0.725706 + 0.688005i \(0.758487\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −11.5174 −0.764440 −0.382220 0.924071i \(-0.624840\pi\)
−0.382220 + 0.924071i \(0.624840\pi\)
\(228\) 0 0
\(229\) 12.8371 0.848300 0.424150 0.905592i \(-0.360573\pi\)
0.424150 + 0.905592i \(0.360573\pi\)
\(230\) 0 0
\(231\) −2.00000 −0.131590
\(232\) 0 0
\(233\) 6.76487 0.443181 0.221591 0.975140i \(-0.428875\pi\)
0.221591 + 0.975140i \(0.428875\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −6.15676 −0.399924
\(238\) 0 0
\(239\) 23.3607 1.51108 0.755539 0.655104i \(-0.227375\pi\)
0.755539 + 0.655104i \(0.227375\pi\)
\(240\) 0 0
\(241\) −14.6803 −0.945644 −0.472822 0.881158i \(-0.656765\pi\)
−0.472822 + 0.881158i \(0.656765\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 2.83710 0.180520
\(248\) 0 0
\(249\) 6.83710 0.433284
\(250\) 0 0
\(251\) −9.16290 −0.578357 −0.289179 0.957275i \(-0.593382\pi\)
−0.289179 + 0.957275i \(0.593382\pi\)
\(252\) 0 0
\(253\) −4.68035 −0.294251
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 5.07838 0.316781 0.158390 0.987377i \(-0.449370\pi\)
0.158390 + 0.987377i \(0.449370\pi\)
\(258\) 0 0
\(259\) −10.8371 −0.673385
\(260\) 0 0
\(261\) −6.68035 −0.413503
\(262\) 0 0
\(263\) 5.65983 0.349000 0.174500 0.984657i \(-0.444169\pi\)
0.174500 + 0.984657i \(0.444169\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 8.34017 0.510410
\(268\) 0 0
\(269\) −27.8576 −1.69851 −0.849255 0.527984i \(-0.822948\pi\)
−0.849255 + 0.527984i \(0.822948\pi\)
\(270\) 0 0
\(271\) −25.1194 −1.52590 −0.762948 0.646460i \(-0.776249\pi\)
−0.762948 + 0.646460i \(0.776249\pi\)
\(272\) 0 0
\(273\) −0.921622 −0.0557791
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 28.1978 1.69424 0.847121 0.531401i \(-0.178334\pi\)
0.847121 + 0.531401i \(0.178334\pi\)
\(278\) 0 0
\(279\) 7.75872 0.464503
\(280\) 0 0
\(281\) −20.3545 −1.21425 −0.607125 0.794606i \(-0.707677\pi\)
−0.607125 + 0.794606i \(0.707677\pi\)
\(282\) 0 0
\(283\) 23.5174 1.39797 0.698984 0.715138i \(-0.253636\pi\)
0.698984 + 0.715138i \(0.253636\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 6.49693 0.383502
\(288\) 0 0
\(289\) −15.8371 −0.931594
\(290\) 0 0
\(291\) 8.43907 0.494707
\(292\) 0 0
\(293\) 2.92162 0.170683 0.0853415 0.996352i \(-0.472802\pi\)
0.0853415 + 0.996352i \(0.472802\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −2.00000 −0.116052
\(298\) 0 0
\(299\) −2.15676 −0.124728
\(300\) 0 0
\(301\) −6.52359 −0.376014
\(302\) 0 0
\(303\) −5.81658 −0.334154
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −10.4703 −0.597570 −0.298785 0.954321i \(-0.596581\pi\)
−0.298785 + 0.954321i \(0.596581\pi\)
\(308\) 0 0
\(309\) −2.15676 −0.122694
\(310\) 0 0
\(311\) −23.8310 −1.35133 −0.675665 0.737209i \(-0.736143\pi\)
−0.675665 + 0.737209i \(0.736143\pi\)
\(312\) 0 0
\(313\) −32.7526 −1.85129 −0.925643 0.378399i \(-0.876475\pi\)
−0.925643 + 0.378399i \(0.876475\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 17.9155 1.00623 0.503117 0.864218i \(-0.332187\pi\)
0.503117 + 0.864218i \(0.332187\pi\)
\(318\) 0 0
\(319\) 13.3607 0.748055
\(320\) 0 0
\(321\) −16.4969 −0.920769
\(322\) 0 0
\(323\) 3.31965 0.184710
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −12.8371 −0.709893
\(328\) 0 0
\(329\) 4.68035 0.258036
\(330\) 0 0
\(331\) 1.36069 0.0747904 0.0373952 0.999301i \(-0.488094\pi\)
0.0373952 + 0.999301i \(0.488094\pi\)
\(332\) 0 0
\(333\) −10.8371 −0.593870
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 25.3607 1.38148 0.690742 0.723101i \(-0.257284\pi\)
0.690742 + 0.723101i \(0.257284\pi\)
\(338\) 0 0
\(339\) 5.23513 0.284333
\(340\) 0 0
\(341\) −15.5174 −0.840317
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 16.8638 0.905294 0.452647 0.891690i \(-0.350480\pi\)
0.452647 + 0.891690i \(0.350480\pi\)
\(348\) 0 0
\(349\) 9.51745 0.509457 0.254729 0.967013i \(-0.418014\pi\)
0.254729 + 0.967013i \(0.418014\pi\)
\(350\) 0 0
\(351\) −0.921622 −0.0491926
\(352\) 0 0
\(353\) 35.7998 1.90543 0.952715 0.303867i \(-0.0982778\pi\)
0.952715 + 0.303867i \(0.0982778\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −1.07838 −0.0570738
\(358\) 0 0
\(359\) 22.3135 1.17766 0.588831 0.808256i \(-0.299588\pi\)
0.588831 + 0.808256i \(0.299588\pi\)
\(360\) 0 0
\(361\) −9.52359 −0.501242
\(362\) 0 0
\(363\) −7.00000 −0.367405
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 20.3135 1.06036 0.530178 0.847886i \(-0.322125\pi\)
0.530178 + 0.847886i \(0.322125\pi\)
\(368\) 0 0
\(369\) 6.49693 0.338217
\(370\) 0 0
\(371\) −3.75872 −0.195143
\(372\) 0 0
\(373\) 16.0000 0.828449 0.414224 0.910175i \(-0.364053\pi\)
0.414224 + 0.910175i \(0.364053\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 6.15676 0.317089
\(378\) 0 0
\(379\) −6.15676 −0.316251 −0.158126 0.987419i \(-0.550545\pi\)
−0.158126 + 0.987419i \(0.550545\pi\)
\(380\) 0 0
\(381\) 1.84324 0.0944323
\(382\) 0 0
\(383\) 26.8371 1.37131 0.685656 0.727926i \(-0.259515\pi\)
0.685656 + 0.727926i \(0.259515\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −6.52359 −0.331613
\(388\) 0 0
\(389\) −5.63317 −0.285613 −0.142806 0.989751i \(-0.545613\pi\)
−0.142806 + 0.989751i \(0.545613\pi\)
\(390\) 0 0
\(391\) −2.52359 −0.127623
\(392\) 0 0
\(393\) −1.47641 −0.0744750
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −37.7998 −1.89712 −0.948558 0.316604i \(-0.897457\pi\)
−0.948558 + 0.316604i \(0.897457\pi\)
\(398\) 0 0
\(399\) −3.07838 −0.154112
\(400\) 0 0
\(401\) −13.6332 −0.680808 −0.340404 0.940279i \(-0.610564\pi\)
−0.340404 + 0.940279i \(0.610564\pi\)
\(402\) 0 0
\(403\) −7.15061 −0.356197
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 21.6742 1.07435
\(408\) 0 0
\(409\) 12.3545 0.610893 0.305447 0.952209i \(-0.401194\pi\)
0.305447 + 0.952209i \(0.401194\pi\)
\(410\) 0 0
\(411\) −4.43907 −0.218963
\(412\) 0 0
\(413\) −10.5236 −0.517832
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −13.6020 −0.666091
\(418\) 0 0
\(419\) −28.9939 −1.41644 −0.708221 0.705991i \(-0.750502\pi\)
−0.708221 + 0.705991i \(0.750502\pi\)
\(420\) 0 0
\(421\) −15.1629 −0.738994 −0.369497 0.929232i \(-0.620470\pi\)
−0.369497 + 0.929232i \(0.620470\pi\)
\(422\) 0 0
\(423\) 4.68035 0.227566
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −4.15676 −0.201159
\(428\) 0 0
\(429\) 1.84324 0.0889927
\(430\) 0 0
\(431\) 10.3135 0.496784 0.248392 0.968660i \(-0.420098\pi\)
0.248392 + 0.968660i \(0.420098\pi\)
\(432\) 0 0
\(433\) −20.4391 −0.982239 −0.491120 0.871092i \(-0.663412\pi\)
−0.491120 + 0.871092i \(0.663412\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −7.20394 −0.344611
\(438\) 0 0
\(439\) 16.9216 0.807625 0.403812 0.914842i \(-0.367685\pi\)
0.403812 + 0.914842i \(0.367685\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −12.8104 −0.608642 −0.304321 0.952569i \(-0.598430\pi\)
−0.304321 + 0.952569i \(0.598430\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 15.6742 0.741364
\(448\) 0 0
\(449\) −14.6270 −0.690292 −0.345146 0.938549i \(-0.612171\pi\)
−0.345146 + 0.938549i \(0.612171\pi\)
\(450\) 0 0
\(451\) −12.9939 −0.611857
\(452\) 0 0
\(453\) −5.84324 −0.274540
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −14.1568 −0.662225 −0.331113 0.943591i \(-0.607424\pi\)
−0.331113 + 0.943591i \(0.607424\pi\)
\(458\) 0 0
\(459\) −1.07838 −0.0503344
\(460\) 0 0
\(461\) 0.340173 0.0158434 0.00792172 0.999969i \(-0.497478\pi\)
0.00792172 + 0.999969i \(0.497478\pi\)
\(462\) 0 0
\(463\) −9.84324 −0.457454 −0.228727 0.973491i \(-0.573456\pi\)
−0.228727 + 0.973491i \(0.573456\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −11.5174 −0.532964 −0.266482 0.963840i \(-0.585861\pi\)
−0.266482 + 0.963840i \(0.585861\pi\)
\(468\) 0 0
\(469\) 4.68035 0.216118
\(470\) 0 0
\(471\) −4.92162 −0.226776
\(472\) 0 0
\(473\) 13.0472 0.599910
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −3.75872 −0.172100
\(478\) 0 0
\(479\) 19.5174 0.891775 0.445887 0.895089i \(-0.352888\pi\)
0.445887 + 0.895089i \(0.352888\pi\)
\(480\) 0 0
\(481\) 9.98771 0.455401
\(482\) 0 0
\(483\) 2.34017 0.106482
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 23.1506 1.04905 0.524527 0.851394i \(-0.324242\pi\)
0.524527 + 0.851394i \(0.324242\pi\)
\(488\) 0 0
\(489\) −9.84324 −0.445127
\(490\) 0 0
\(491\) −2.00000 −0.0902587 −0.0451294 0.998981i \(-0.514370\pi\)
−0.0451294 + 0.998981i \(0.514370\pi\)
\(492\) 0 0
\(493\) 7.20394 0.324449
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −2.00000 −0.0897123
\(498\) 0 0
\(499\) −27.2039 −1.21782 −0.608908 0.793241i \(-0.708392\pi\)
−0.608908 + 0.793241i \(0.708392\pi\)
\(500\) 0 0
\(501\) −19.2039 −0.857969
\(502\) 0 0
\(503\) −18.8371 −0.839905 −0.419952 0.907546i \(-0.637953\pi\)
−0.419952 + 0.907546i \(0.637953\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −12.1506 −0.539628
\(508\) 0 0
\(509\) 6.81044 0.301867 0.150934 0.988544i \(-0.451772\pi\)
0.150934 + 0.988544i \(0.451772\pi\)
\(510\) 0 0
\(511\) −7.07838 −0.313129
\(512\) 0 0
\(513\) −3.07838 −0.135914
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −9.36069 −0.411683
\(518\) 0 0
\(519\) 22.4391 0.984966
\(520\) 0 0
\(521\) −25.8166 −1.13105 −0.565523 0.824733i \(-0.691325\pi\)
−0.565523 + 0.824733i \(0.691325\pi\)
\(522\) 0 0
\(523\) 4.00000 0.174908 0.0874539 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −8.36683 −0.364465
\(528\) 0 0
\(529\) −17.5236 −0.761895
\(530\) 0 0
\(531\) −10.5236 −0.456685
\(532\) 0 0
\(533\) −5.98771 −0.259357
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −10.0000 −0.431532
\(538\) 0 0
\(539\) −2.00000 −0.0861461
\(540\) 0 0
\(541\) 25.8843 1.11285 0.556426 0.830897i \(-0.312172\pi\)
0.556426 + 0.830897i \(0.312172\pi\)
\(542\) 0 0
\(543\) −8.52359 −0.365782
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 11.3197 0.483993 0.241997 0.970277i \(-0.422198\pi\)
0.241997 + 0.970277i \(0.422198\pi\)
\(548\) 0 0
\(549\) −4.15676 −0.177406
\(550\) 0 0
\(551\) 20.5646 0.876083
\(552\) 0 0
\(553\) −6.15676 −0.261812
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 26.6491 1.12916 0.564580 0.825378i \(-0.309038\pi\)
0.564580 + 0.825378i \(0.309038\pi\)
\(558\) 0 0
\(559\) 6.01229 0.254293
\(560\) 0 0
\(561\) 2.15676 0.0910583
\(562\) 0 0
\(563\) 46.3545 1.95361 0.976806 0.214128i \(-0.0686909\pi\)
0.976806 + 0.214128i \(0.0686909\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −14.3668 −0.602289 −0.301145 0.953579i \(-0.597369\pi\)
−0.301145 + 0.953579i \(0.597369\pi\)
\(570\) 0 0
\(571\) −38.7214 −1.62044 −0.810220 0.586126i \(-0.800652\pi\)
−0.810220 + 0.586126i \(0.800652\pi\)
\(572\) 0 0
\(573\) −15.3607 −0.641702
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −43.4740 −1.80984 −0.904922 0.425577i \(-0.860071\pi\)
−0.904922 + 0.425577i \(0.860071\pi\)
\(578\) 0 0
\(579\) −8.36683 −0.347714
\(580\) 0 0
\(581\) 6.83710 0.283651
\(582\) 0 0
\(583\) 7.51745 0.311341
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 36.0288 1.48707 0.743533 0.668699i \(-0.233149\pi\)
0.743533 + 0.668699i \(0.233149\pi\)
\(588\) 0 0
\(589\) −23.8843 −0.984135
\(590\) 0 0
\(591\) 11.7587 0.483689
\(592\) 0 0
\(593\) −31.4863 −1.29299 −0.646493 0.762920i \(-0.723765\pi\)
−0.646493 + 0.762920i \(0.723765\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 22.5958 0.924786
\(598\) 0 0
\(599\) −29.0349 −1.18633 −0.593167 0.805080i \(-0.702123\pi\)
−0.593167 + 0.805080i \(0.702123\pi\)
\(600\) 0 0
\(601\) 15.3607 0.626576 0.313288 0.949658i \(-0.398570\pi\)
0.313288 + 0.949658i \(0.398570\pi\)
\(602\) 0 0
\(603\) 4.68035 0.190598
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −13.0472 −0.529569 −0.264784 0.964308i \(-0.585301\pi\)
−0.264784 + 0.964308i \(0.585301\pi\)
\(608\) 0 0
\(609\) −6.68035 −0.270701
\(610\) 0 0
\(611\) −4.31351 −0.174506
\(612\) 0 0
\(613\) 15.5174 0.626744 0.313372 0.949630i \(-0.398541\pi\)
0.313372 + 0.949630i \(0.398541\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 22.7649 0.916479 0.458240 0.888829i \(-0.348480\pi\)
0.458240 + 0.888829i \(0.348480\pi\)
\(618\) 0 0
\(619\) −7.92777 −0.318644 −0.159322 0.987227i \(-0.550931\pi\)
−0.159322 + 0.987227i \(0.550931\pi\)
\(620\) 0 0
\(621\) 2.34017 0.0939079
\(622\) 0 0
\(623\) 8.34017 0.334142
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 6.15676 0.245877
\(628\) 0 0
\(629\) 11.6865 0.465971
\(630\) 0 0
\(631\) −19.2039 −0.764497 −0.382248 0.924060i \(-0.624850\pi\)
−0.382248 + 0.924060i \(0.624850\pi\)
\(632\) 0 0
\(633\) 13.6742 0.543501
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −0.921622 −0.0365160
\(638\) 0 0
\(639\) −2.00000 −0.0791188
\(640\) 0 0
\(641\) −5.94668 −0.234880 −0.117440 0.993080i \(-0.537469\pi\)
−0.117440 + 0.993080i \(0.537469\pi\)
\(642\) 0 0
\(643\) −30.8904 −1.21820 −0.609100 0.793094i \(-0.708469\pi\)
−0.609100 + 0.793094i \(0.708469\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −19.2039 −0.754985 −0.377492 0.926013i \(-0.623214\pi\)
−0.377492 + 0.926013i \(0.623214\pi\)
\(648\) 0 0
\(649\) 21.0472 0.826174
\(650\) 0 0
\(651\) 7.75872 0.304088
\(652\) 0 0
\(653\) 28.5548 1.11744 0.558718 0.829358i \(-0.311294\pi\)
0.558718 + 0.829358i \(0.311294\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −7.07838 −0.276154
\(658\) 0 0
\(659\) 27.9877 1.09025 0.545123 0.838356i \(-0.316483\pi\)
0.545123 + 0.838356i \(0.316483\pi\)
\(660\) 0 0
\(661\) −22.1445 −0.861320 −0.430660 0.902514i \(-0.641719\pi\)
−0.430660 + 0.902514i \(0.641719\pi\)
\(662\) 0 0
\(663\) 0.993857 0.0385982
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −15.6332 −0.605319
\(668\) 0 0
\(669\) −21.6742 −0.837973
\(670\) 0 0
\(671\) 8.31351 0.320940
\(672\) 0 0
\(673\) −2.21008 −0.0851923 −0.0425962 0.999092i \(-0.513563\pi\)
−0.0425962 + 0.999092i \(0.513563\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −19.5486 −0.751315 −0.375658 0.926758i \(-0.622583\pi\)
−0.375658 + 0.926758i \(0.622583\pi\)
\(678\) 0 0
\(679\) 8.43907 0.323862
\(680\) 0 0
\(681\) −11.5174 −0.441350
\(682\) 0 0
\(683\) 11.8166 0.452149 0.226074 0.974110i \(-0.427411\pi\)
0.226074 + 0.974110i \(0.427411\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 12.8371 0.489766
\(688\) 0 0
\(689\) 3.46412 0.131973
\(690\) 0 0
\(691\) −11.7587 −0.447323 −0.223661 0.974667i \(-0.571801\pi\)
−0.223661 + 0.974667i \(0.571801\pi\)
\(692\) 0 0
\(693\) −2.00000 −0.0759737
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −7.00614 −0.265377
\(698\) 0 0
\(699\) 6.76487 0.255871
\(700\) 0 0
\(701\) 9.94668 0.375681 0.187840 0.982200i \(-0.439851\pi\)
0.187840 + 0.982200i \(0.439851\pi\)
\(702\) 0 0
\(703\) 33.3607 1.25822
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −5.81658 −0.218755
\(708\) 0 0
\(709\) −11.0472 −0.414886 −0.207443 0.978247i \(-0.566514\pi\)
−0.207443 + 0.978247i \(0.566514\pi\)
\(710\) 0 0
\(711\) −6.15676 −0.230896
\(712\) 0 0
\(713\) 18.1568 0.679976
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 23.3607 0.872421
\(718\) 0 0
\(719\) 6.15676 0.229608 0.114804 0.993388i \(-0.463376\pi\)
0.114804 + 0.993388i \(0.463376\pi\)
\(720\) 0 0
\(721\) −2.15676 −0.0803218
\(722\) 0 0
\(723\) −14.6803 −0.545968
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 2.89043 0.107200 0.0536000 0.998562i \(-0.482930\pi\)
0.0536000 + 0.998562i \(0.482930\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 7.03489 0.260195
\(732\) 0 0
\(733\) 25.7998 0.952936 0.476468 0.879192i \(-0.341917\pi\)
0.476468 + 0.879192i \(0.341917\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −9.36069 −0.344806
\(738\) 0 0
\(739\) 1.04718 0.0385212 0.0192606 0.999814i \(-0.493869\pi\)
0.0192606 + 0.999814i \(0.493869\pi\)
\(740\) 0 0
\(741\) 2.83710 0.104224
\(742\) 0 0
\(743\) −9.97334 −0.365886 −0.182943 0.983123i \(-0.558562\pi\)
−0.182943 + 0.983123i \(0.558562\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 6.83710 0.250156
\(748\) 0 0
\(749\) −16.4969 −0.602785
\(750\) 0 0
\(751\) −3.26633 −0.119190 −0.0595950 0.998223i \(-0.518981\pi\)
−0.0595950 + 0.998223i \(0.518981\pi\)
\(752\) 0 0
\(753\) −9.16290 −0.333915
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 49.9877 1.81683 0.908417 0.418065i \(-0.137292\pi\)
0.908417 + 0.418065i \(0.137292\pi\)
\(758\) 0 0
\(759\) −4.68035 −0.169886
\(760\) 0 0
\(761\) 2.61265 0.0947083 0.0473542 0.998878i \(-0.484921\pi\)
0.0473542 + 0.998878i \(0.484921\pi\)
\(762\) 0 0
\(763\) −12.8371 −0.464734
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 9.69878 0.350202
\(768\) 0 0
\(769\) −15.6742 −0.565226 −0.282613 0.959234i \(-0.591201\pi\)
−0.282613 + 0.959234i \(0.591201\pi\)
\(770\) 0 0
\(771\) 5.07838 0.182893
\(772\) 0 0
\(773\) −5.81205 −0.209045 −0.104522 0.994523i \(-0.533331\pi\)
−0.104522 + 0.994523i \(0.533331\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −10.8371 −0.388779
\(778\) 0 0
\(779\) −20.0000 −0.716574
\(780\) 0 0
\(781\) 4.00000 0.143131
\(782\) 0 0
\(783\) −6.68035 −0.238736
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 39.3484 1.40262 0.701310 0.712857i \(-0.252599\pi\)
0.701310 + 0.712857i \(0.252599\pi\)
\(788\) 0 0
\(789\) 5.65983 0.201495
\(790\) 0 0
\(791\) 5.23513 0.186140
\(792\) 0 0
\(793\) 3.83096 0.136041
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −28.2823 −1.00181 −0.500905 0.865502i \(-0.667000\pi\)
−0.500905 + 0.865502i \(0.667000\pi\)
\(798\) 0 0
\(799\) −5.04718 −0.178556
\(800\) 0 0
\(801\) 8.34017 0.294686
\(802\) 0 0
\(803\) 14.1568 0.499581
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −27.8576 −0.980635
\(808\) 0 0
\(809\) −15.6742 −0.551076 −0.275538 0.961290i \(-0.588856\pi\)
−0.275538 + 0.961290i \(0.588856\pi\)
\(810\) 0 0
\(811\) 42.1666 1.48067 0.740335 0.672238i \(-0.234667\pi\)
0.740335 + 0.672238i \(0.234667\pi\)
\(812\) 0 0
\(813\) −25.1194 −0.880976
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 20.0821 0.702583
\(818\) 0 0
\(819\) −0.921622 −0.0322041
\(820\) 0 0
\(821\) −39.0472 −1.36276 −0.681378 0.731932i \(-0.738619\pi\)
−0.681378 + 0.731932i \(0.738619\pi\)
\(822\) 0 0
\(823\) −36.5646 −1.27456 −0.637281 0.770631i \(-0.719941\pi\)
−0.637281 + 0.770631i \(0.719941\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 50.2245 1.74648 0.873238 0.487294i \(-0.162016\pi\)
0.873238 + 0.487294i \(0.162016\pi\)
\(828\) 0 0
\(829\) 32.8371 1.14048 0.570240 0.821478i \(-0.306850\pi\)
0.570240 + 0.821478i \(0.306850\pi\)
\(830\) 0 0
\(831\) 28.1978 0.978171
\(832\) 0 0
\(833\) −1.07838 −0.0373636
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 7.75872 0.268181
\(838\) 0 0
\(839\) 13.3607 0.461262 0.230631 0.973041i \(-0.425921\pi\)
0.230631 + 0.973041i \(0.425921\pi\)
\(840\) 0 0
\(841\) 15.6270 0.538863
\(842\) 0 0
\(843\) −20.3545 −0.701048
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −7.00000 −0.240523
\(848\) 0 0
\(849\) 23.5174 0.807117
\(850\) 0 0
\(851\) −25.3607 −0.869353
\(852\) 0 0
\(853\) 39.6430 1.35735 0.678675 0.734438i \(-0.262554\pi\)
0.678675 + 0.734438i \(0.262554\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 29.7054 1.01472 0.507359 0.861735i \(-0.330622\pi\)
0.507359 + 0.861735i \(0.330622\pi\)
\(858\) 0 0
\(859\) 3.07838 0.105033 0.0525164 0.998620i \(-0.483276\pi\)
0.0525164 + 0.998620i \(0.483276\pi\)
\(860\) 0 0
\(861\) 6.49693 0.221415
\(862\) 0 0
\(863\) −6.39350 −0.217637 −0.108819 0.994062i \(-0.534707\pi\)
−0.108819 + 0.994062i \(0.534707\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −15.8371 −0.537856
\(868\) 0 0
\(869\) 12.3135 0.417707
\(870\) 0 0
\(871\) −4.31351 −0.146158
\(872\) 0 0
\(873\) 8.43907 0.285619
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1.21622 0.0410689 0.0205345 0.999789i \(-0.493463\pi\)
0.0205345 + 0.999789i \(0.493463\pi\)
\(878\) 0 0
\(879\) 2.92162 0.0985439
\(880\) 0 0
\(881\) 15.9733 0.538155 0.269078 0.963118i \(-0.413281\pi\)
0.269078 + 0.963118i \(0.413281\pi\)
\(882\) 0 0
\(883\) 11.6865 0.393282 0.196641 0.980476i \(-0.436997\pi\)
0.196641 + 0.980476i \(0.436997\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −25.6209 −0.860265 −0.430132 0.902766i \(-0.641533\pi\)
−0.430132 + 0.902766i \(0.641533\pi\)
\(888\) 0 0
\(889\) 1.84324 0.0618204
\(890\) 0 0
\(891\) −2.00000 −0.0670025
\(892\) 0 0
\(893\) −14.4079 −0.482141
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −2.15676 −0.0720120
\(898\) 0 0
\(899\) −51.8310 −1.72866
\(900\) 0 0
\(901\) 4.05332 0.135036
\(902\) 0 0
\(903\) −6.52359 −0.217091
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −57.7563 −1.91777 −0.958883 0.283802i \(-0.908404\pi\)
−0.958883 + 0.283802i \(0.908404\pi\)
\(908\) 0 0
\(909\) −5.81658 −0.192924
\(910\) 0 0
\(911\) 35.9877 1.19233 0.596163 0.802863i \(-0.296691\pi\)
0.596163 + 0.802863i \(0.296691\pi\)
\(912\) 0 0
\(913\) −13.6742 −0.452550
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −1.47641 −0.0487553
\(918\) 0 0
\(919\) −46.7214 −1.54120 −0.770598 0.637321i \(-0.780042\pi\)
−0.770598 + 0.637321i \(0.780042\pi\)
\(920\) 0 0
\(921\) −10.4703 −0.345007
\(922\) 0 0
\(923\) 1.84324 0.0606711
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −2.15676 −0.0708371
\(928\) 0 0
\(929\) −53.0493 −1.74049 −0.870245 0.492619i \(-0.836040\pi\)
−0.870245 + 0.492619i \(0.836040\pi\)
\(930\) 0 0
\(931\) −3.07838 −0.100890
\(932\) 0 0
\(933\) −23.8310 −0.780191
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 16.1256 0.526799 0.263400 0.964687i \(-0.415156\pi\)
0.263400 + 0.964687i \(0.415156\pi\)
\(938\) 0 0
\(939\) −32.7526 −1.06884
\(940\) 0 0
\(941\) 24.7070 0.805425 0.402713 0.915326i \(-0.368067\pi\)
0.402713 + 0.915326i \(0.368067\pi\)
\(942\) 0 0
\(943\) 15.2039 0.495108
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −6.53797 −0.212455 −0.106228 0.994342i \(-0.533877\pi\)
−0.106228 + 0.994342i \(0.533877\pi\)
\(948\) 0 0
\(949\) 6.52359 0.211765
\(950\) 0 0
\(951\) 17.9155 0.580949
\(952\) 0 0
\(953\) −6.11327 −0.198028 −0.0990142 0.995086i \(-0.531569\pi\)
−0.0990142 + 0.995086i \(0.531569\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 13.3607 0.431890
\(958\) 0 0
\(959\) −4.43907 −0.143345
\(960\) 0 0
\(961\) 29.1978 0.941864
\(962\) 0 0
\(963\) −16.4969 −0.531606
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −25.6209 −0.823912 −0.411956 0.911204i \(-0.635154\pi\)
−0.411956 + 0.911204i \(0.635154\pi\)
\(968\) 0 0
\(969\) 3.31965 0.106643
\(970\) 0 0
\(971\) −4.05332 −0.130077 −0.0650387 0.997883i \(-0.520717\pi\)
−0.0650387 + 0.997883i \(0.520717\pi\)
\(972\) 0 0
\(973\) −13.6020 −0.436059
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 3.81205 0.121958 0.0609791 0.998139i \(-0.480578\pi\)
0.0609791 + 0.998139i \(0.480578\pi\)
\(978\) 0 0
\(979\) −16.6803 −0.533106
\(980\) 0 0
\(981\) −12.8371 −0.409857
\(982\) 0 0
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 4.68035 0.148977
\(988\) 0 0
\(989\) −15.2663 −0.485441
\(990\) 0 0
\(991\) 42.4079 1.34713 0.673565 0.739128i \(-0.264762\pi\)
0.673565 + 0.739128i \(0.264762\pi\)
\(992\) 0 0
\(993\) 1.36069 0.0431803
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −43.4740 −1.37683 −0.688417 0.725315i \(-0.741694\pi\)
−0.688417 + 0.725315i \(0.741694\pi\)
\(998\) 0 0
\(999\) −10.8371 −0.342871
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.dj.1.2 3
4.3 odd 2 525.2.a.k.1.3 3
5.2 odd 4 1680.2.t.k.1009.3 6
5.3 odd 4 1680.2.t.k.1009.6 6
5.4 even 2 8400.2.a.dg.1.2 3
12.11 even 2 1575.2.a.w.1.1 3
15.2 even 4 5040.2.t.v.1009.1 6
15.8 even 4 5040.2.t.v.1009.2 6
20.3 even 4 105.2.d.b.64.1 6
20.7 even 4 105.2.d.b.64.6 yes 6
20.19 odd 2 525.2.a.j.1.1 3
28.27 even 2 3675.2.a.bj.1.3 3
60.23 odd 4 315.2.d.e.64.6 6
60.47 odd 4 315.2.d.e.64.1 6
60.59 even 2 1575.2.a.x.1.3 3
140.3 odd 12 735.2.q.f.79.6 12
140.23 even 12 735.2.q.e.214.1 12
140.27 odd 4 735.2.d.b.589.6 6
140.47 odd 12 735.2.q.f.214.6 12
140.67 even 12 735.2.q.e.79.1 12
140.83 odd 4 735.2.d.b.589.1 6
140.87 odd 12 735.2.q.f.79.1 12
140.103 odd 12 735.2.q.f.214.1 12
140.107 even 12 735.2.q.e.214.6 12
140.123 even 12 735.2.q.e.79.6 12
140.139 even 2 3675.2.a.bi.1.1 3
420.83 even 4 2205.2.d.l.1324.6 6
420.167 even 4 2205.2.d.l.1324.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.2.d.b.64.1 6 20.3 even 4
105.2.d.b.64.6 yes 6 20.7 even 4
315.2.d.e.64.1 6 60.47 odd 4
315.2.d.e.64.6 6 60.23 odd 4
525.2.a.j.1.1 3 20.19 odd 2
525.2.a.k.1.3 3 4.3 odd 2
735.2.d.b.589.1 6 140.83 odd 4
735.2.d.b.589.6 6 140.27 odd 4
735.2.q.e.79.1 12 140.67 even 12
735.2.q.e.79.6 12 140.123 even 12
735.2.q.e.214.1 12 140.23 even 12
735.2.q.e.214.6 12 140.107 even 12
735.2.q.f.79.1 12 140.87 odd 12
735.2.q.f.79.6 12 140.3 odd 12
735.2.q.f.214.1 12 140.103 odd 12
735.2.q.f.214.6 12 140.47 odd 12
1575.2.a.w.1.1 3 12.11 even 2
1575.2.a.x.1.3 3 60.59 even 2
1680.2.t.k.1009.3 6 5.2 odd 4
1680.2.t.k.1009.6 6 5.3 odd 4
2205.2.d.l.1324.1 6 420.167 even 4
2205.2.d.l.1324.6 6 420.83 even 4
3675.2.a.bi.1.1 3 140.139 even 2
3675.2.a.bj.1.3 3 28.27 even 2
5040.2.t.v.1009.1 6 15.2 even 4
5040.2.t.v.1009.2 6 15.8 even 4
8400.2.a.dg.1.2 3 5.4 even 2
8400.2.a.dj.1.2 3 1.1 even 1 trivial