Properties

Label 6400.2.a.bg.1.1
Level $6400$
Weight $2$
Character 6400.1
Self dual yes
Analytic conductor $51.104$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 6400 = 2^{8} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6400.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(51.1042572936\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
Defining polynomial: \( x^{2} - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 200)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.64575\) of defining polynomial
Character \(\chi\) \(=\) 6400.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.64575 q^{3} -4.00000 q^{7} +4.00000 q^{9} +O(q^{10})\) \(q-2.64575 q^{3} -4.00000 q^{7} +4.00000 q^{9} -2.64575 q^{11} -3.00000 q^{17} +2.64575 q^{19} +10.5830 q^{21} -4.00000 q^{23} -2.64575 q^{27} +4.00000 q^{31} +7.00000 q^{33} +10.5830 q^{37} +5.00000 q^{41} +5.29150 q^{43} -8.00000 q^{47} +9.00000 q^{49} +7.93725 q^{51} -10.5830 q^{53} -7.00000 q^{57} -5.29150 q^{59} +10.5830 q^{61} -16.0000 q^{63} +7.93725 q^{67} +10.5830 q^{69} -8.00000 q^{71} +7.00000 q^{73} +10.5830 q^{77} +4.00000 q^{79} -5.00000 q^{81} +7.93725 q^{83} +1.00000 q^{89} -10.5830 q^{93} -2.00000 q^{97} -10.5830 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{7} + 8 q^{9} - 6 q^{17} - 8 q^{23} + 8 q^{31} + 14 q^{33} + 10 q^{41} - 16 q^{47} + 18 q^{49} - 14 q^{57} - 32 q^{63} - 16 q^{71} + 14 q^{73} + 8 q^{79} - 10 q^{81} + 2 q^{89} - 4 q^{97}+O(q^{100}) \) 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
\(3\) −2.64575 −1.52753 −0.763763 0.645497i \(-0.776650\pi\)
−0.763763 + 0.645497i \(0.776650\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −4.00000 −1.51186 −0.755929 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 0 0
\(9\) 4.00000 1.33333
\(10\) 0 0
\(11\) −2.64575 −0.797724 −0.398862 0.917011i \(-0.630595\pi\)
−0.398862 + 0.917011i \(0.630595\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) 2.64575 0.606977 0.303488 0.952835i \(-0.401849\pi\)
0.303488 + 0.952835i \(0.401849\pi\)
\(20\) 0 0
\(21\) 10.5830 2.30940
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −2.64575 −0.509175
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) 7.00000 1.21854
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 10.5830 1.73984 0.869918 0.493197i \(-0.164172\pi\)
0.869918 + 0.493197i \(0.164172\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.00000 0.780869 0.390434 0.920631i \(-0.372325\pi\)
0.390434 + 0.920631i \(0.372325\pi\)
\(42\) 0 0
\(43\) 5.29150 0.806947 0.403473 0.914991i \(-0.367803\pi\)
0.403473 + 0.914991i \(0.367803\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) 7.93725 1.11144
\(52\) 0 0
\(53\) −10.5830 −1.45369 −0.726844 0.686803i \(-0.759014\pi\)
−0.726844 + 0.686803i \(0.759014\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −7.00000 −0.927173
\(58\) 0 0
\(59\) −5.29150 −0.688895 −0.344447 0.938806i \(-0.611934\pi\)
−0.344447 + 0.938806i \(0.611934\pi\)
\(60\) 0 0
\(61\) 10.5830 1.35501 0.677507 0.735516i \(-0.263060\pi\)
0.677507 + 0.735516i \(0.263060\pi\)
\(62\) 0 0
\(63\) −16.0000 −2.01581
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 7.93725 0.969690 0.484845 0.874600i \(-0.338876\pi\)
0.484845 + 0.874600i \(0.338876\pi\)
\(68\) 0 0
\(69\) 10.5830 1.27404
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 7.00000 0.819288 0.409644 0.912245i \(-0.365653\pi\)
0.409644 + 0.912245i \(0.365653\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 10.5830 1.20605
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) −5.00000 −0.555556
\(82\) 0 0
\(83\) 7.93725 0.871227 0.435613 0.900134i \(-0.356531\pi\)
0.435613 + 0.900134i \(0.356531\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.00000 0.106000 0.0529999 0.998595i \(-0.483122\pi\)
0.0529999 + 0.998595i \(0.483122\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −10.5830 −1.09741
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) −10.5830 −1.06363
\(100\) 0 0
\(101\) 10.5830 1.05305 0.526524 0.850160i \(-0.323495\pi\)
0.526524 + 0.850160i \(0.323495\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.64575 0.255774 0.127887 0.991789i \(-0.459180\pi\)
0.127887 + 0.991789i \(0.459180\pi\)
\(108\) 0 0
\(109\) 10.5830 1.01367 0.506834 0.862044i \(-0.330816\pi\)
0.506834 + 0.862044i \(0.330816\pi\)
\(110\) 0 0
\(111\) −28.0000 −2.65764
\(112\) 0 0
\(113\) −15.0000 −1.41108 −0.705541 0.708669i \(-0.749296\pi\)
−0.705541 + 0.708669i \(0.749296\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 12.0000 1.10004
\(120\) 0 0
\(121\) −4.00000 −0.363636
\(122\) 0 0
\(123\) −13.2288 −1.19280
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 12.0000 1.06483 0.532414 0.846484i \(-0.321285\pi\)
0.532414 + 0.846484i \(0.321285\pi\)
\(128\) 0 0
\(129\) −14.0000 −1.23263
\(130\) 0 0
\(131\) −15.8745 −1.38696 −0.693481 0.720475i \(-0.743924\pi\)
−0.693481 + 0.720475i \(0.743924\pi\)
\(132\) 0 0
\(133\) −10.5830 −0.917663
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 19.0000 1.62328 0.811640 0.584158i \(-0.198575\pi\)
0.811640 + 0.584158i \(0.198575\pi\)
\(138\) 0 0
\(139\) 18.5203 1.57087 0.785434 0.618945i \(-0.212440\pi\)
0.785434 + 0.618945i \(0.212440\pi\)
\(140\) 0 0
\(141\) 21.1660 1.78250
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −23.8118 −1.96396
\(148\) 0 0
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) 0 0
\(153\) −12.0000 −0.970143
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 10.5830 0.844616 0.422308 0.906452i \(-0.361220\pi\)
0.422308 + 0.906452i \(0.361220\pi\)
\(158\) 0 0
\(159\) 28.0000 2.22054
\(160\) 0 0
\(161\) 16.0000 1.26098
\(162\) 0 0
\(163\) −13.2288 −1.03616 −0.518078 0.855333i \(-0.673352\pi\)
−0.518078 + 0.855333i \(0.673352\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 10.5830 0.809303
\(172\) 0 0
\(173\) −21.1660 −1.60922 −0.804611 0.593802i \(-0.797626\pi\)
−0.804611 + 0.593802i \(0.797626\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 14.0000 1.05230
\(178\) 0 0
\(179\) 23.8118 1.77977 0.889887 0.456180i \(-0.150783\pi\)
0.889887 + 0.456180i \(0.150783\pi\)
\(180\) 0 0
\(181\) −10.5830 −0.786629 −0.393314 0.919404i \(-0.628672\pi\)
−0.393314 + 0.919404i \(0.628672\pi\)
\(182\) 0 0
\(183\) −28.0000 −2.06982
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 7.93725 0.580429
\(188\) 0 0
\(189\) 10.5830 0.769800
\(190\) 0 0
\(191\) 4.00000 0.289430 0.144715 0.989473i \(-0.453773\pi\)
0.144715 + 0.989473i \(0.453773\pi\)
\(192\) 0 0
\(193\) 5.00000 0.359908 0.179954 0.983675i \(-0.442405\pi\)
0.179954 + 0.983675i \(0.442405\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −10.5830 −0.754008 −0.377004 0.926212i \(-0.623046\pi\)
−0.377004 + 0.926212i \(0.623046\pi\)
\(198\) 0 0
\(199\) 24.0000 1.70131 0.850657 0.525720i \(-0.176204\pi\)
0.850657 + 0.525720i \(0.176204\pi\)
\(200\) 0 0
\(201\) −21.0000 −1.48123
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −16.0000 −1.11208
\(208\) 0 0
\(209\) −7.00000 −0.484200
\(210\) 0 0
\(211\) −7.93725 −0.546423 −0.273212 0.961954i \(-0.588086\pi\)
−0.273212 + 0.961954i \(0.588086\pi\)
\(212\) 0 0
\(213\) 21.1660 1.45027
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −16.0000 −1.08615
\(218\) 0 0
\(219\) −18.5203 −1.25148
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 15.8745 1.05363 0.526814 0.849981i \(-0.323386\pi\)
0.526814 + 0.849981i \(0.323386\pi\)
\(228\) 0 0
\(229\) −21.1660 −1.39869 −0.699345 0.714785i \(-0.746525\pi\)
−0.699345 + 0.714785i \(0.746525\pi\)
\(230\) 0 0
\(231\) −28.0000 −1.84226
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −10.5830 −0.687440
\(238\) 0 0
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) −21.0000 −1.35273 −0.676364 0.736567i \(-0.736446\pi\)
−0.676364 + 0.736567i \(0.736446\pi\)
\(242\) 0 0
\(243\) 21.1660 1.35780
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −21.0000 −1.33082
\(250\) 0 0
\(251\) 7.93725 0.500995 0.250498 0.968117i \(-0.419406\pi\)
0.250498 + 0.968117i \(0.419406\pi\)
\(252\) 0 0
\(253\) 10.5830 0.665348
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 14.0000 0.873296 0.436648 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) 0 0
\(259\) −42.3320 −2.63038
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −12.0000 −0.739952 −0.369976 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −2.64575 −0.161917
\(268\) 0 0
\(269\) −21.1660 −1.29051 −0.645257 0.763965i \(-0.723250\pi\)
−0.645257 + 0.763965i \(0.723250\pi\)
\(270\) 0 0
\(271\) −20.0000 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 21.1660 1.27174 0.635871 0.771795i \(-0.280641\pi\)
0.635871 + 0.771795i \(0.280641\pi\)
\(278\) 0 0
\(279\) 16.0000 0.957895
\(280\) 0 0
\(281\) 22.0000 1.31241 0.656205 0.754583i \(-0.272161\pi\)
0.656205 + 0.754583i \(0.272161\pi\)
\(282\) 0 0
\(283\) 13.2288 0.786368 0.393184 0.919460i \(-0.371374\pi\)
0.393184 + 0.919460i \(0.371374\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −20.0000 −1.18056
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 5.29150 0.310193
\(292\) 0 0
\(293\) 10.5830 0.618266 0.309133 0.951019i \(-0.399961\pi\)
0.309133 + 0.951019i \(0.399961\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 7.00000 0.406181
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −21.1660 −1.21999
\(302\) 0 0
\(303\) −28.0000 −1.60856
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −2.64575 −0.151001 −0.0755005 0.997146i \(-0.524055\pi\)
−0.0755005 + 0.997146i \(0.524055\pi\)
\(308\) 0 0
\(309\) 21.1660 1.20409
\(310\) 0 0
\(311\) 4.00000 0.226819 0.113410 0.993548i \(-0.463823\pi\)
0.113410 + 0.993548i \(0.463823\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −7.00000 −0.390702
\(322\) 0 0
\(323\) −7.93725 −0.441641
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −28.0000 −1.54840
\(328\) 0 0
\(329\) 32.0000 1.76422
\(330\) 0 0
\(331\) −2.64575 −0.145424 −0.0727118 0.997353i \(-0.523165\pi\)
−0.0727118 + 0.997353i \(0.523165\pi\)
\(332\) 0 0
\(333\) 42.3320 2.31978
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −15.0000 −0.817102 −0.408551 0.912735i \(-0.633966\pi\)
−0.408551 + 0.912735i \(0.633966\pi\)
\(338\) 0 0
\(339\) 39.6863 2.15546
\(340\) 0 0
\(341\) −10.5830 −0.573102
\(342\) 0 0
\(343\) −8.00000 −0.431959
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 2.64575 0.142031 0.0710157 0.997475i \(-0.477376\pi\)
0.0710157 + 0.997475i \(0.477376\pi\)
\(348\) 0 0
\(349\) −10.5830 −0.566495 −0.283248 0.959047i \(-0.591412\pi\)
−0.283248 + 0.959047i \(0.591412\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −31.7490 −1.68034
\(358\) 0 0
\(359\) −36.0000 −1.90001 −0.950004 0.312239i \(-0.898921\pi\)
−0.950004 + 0.312239i \(0.898921\pi\)
\(360\) 0 0
\(361\) −12.0000 −0.631579
\(362\) 0 0
\(363\) 10.5830 0.555464
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) 20.0000 1.04116
\(370\) 0 0
\(371\) 42.3320 2.19777
\(372\) 0 0
\(373\) −10.5830 −0.547967 −0.273984 0.961734i \(-0.588341\pi\)
−0.273984 + 0.961734i \(0.588341\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 7.93725 0.407709 0.203855 0.979001i \(-0.434653\pi\)
0.203855 + 0.979001i \(0.434653\pi\)
\(380\) 0 0
\(381\) −31.7490 −1.62655
\(382\) 0 0
\(383\) −36.0000 −1.83951 −0.919757 0.392488i \(-0.871614\pi\)
−0.919757 + 0.392488i \(0.871614\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 21.1660 1.07593
\(388\) 0 0
\(389\) −10.5830 −0.536580 −0.268290 0.963338i \(-0.586458\pi\)
−0.268290 + 0.963338i \(0.586458\pi\)
\(390\) 0 0
\(391\) 12.0000 0.606866
\(392\) 0 0
\(393\) 42.0000 2.11862
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −21.1660 −1.06229 −0.531146 0.847280i \(-0.678238\pi\)
−0.531146 + 0.847280i \(0.678238\pi\)
\(398\) 0 0
\(399\) 28.0000 1.40175
\(400\) 0 0
\(401\) 27.0000 1.34832 0.674158 0.738587i \(-0.264507\pi\)
0.674158 + 0.738587i \(0.264507\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −28.0000 −1.38791
\(408\) 0 0
\(409\) −3.00000 −0.148340 −0.0741702 0.997246i \(-0.523631\pi\)
−0.0741702 + 0.997246i \(0.523631\pi\)
\(410\) 0 0
\(411\) −50.2693 −2.47960
\(412\) 0 0
\(413\) 21.1660 1.04151
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −49.0000 −2.39954
\(418\) 0 0
\(419\) −18.5203 −0.904774 −0.452387 0.891822i \(-0.649427\pi\)
−0.452387 + 0.891822i \(0.649427\pi\)
\(420\) 0 0
\(421\) −21.1660 −1.03157 −0.515784 0.856719i \(-0.672499\pi\)
−0.515784 + 0.856719i \(0.672499\pi\)
\(422\) 0 0
\(423\) −32.0000 −1.55589
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −42.3320 −2.04859
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 12.0000 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) 0 0
\(433\) 37.0000 1.77811 0.889053 0.457804i \(-0.151364\pi\)
0.889053 + 0.457804i \(0.151364\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −10.5830 −0.506254
\(438\) 0 0
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 0 0
\(441\) 36.0000 1.71429
\(442\) 0 0
\(443\) −29.1033 −1.38274 −0.691369 0.722502i \(-0.742992\pi\)
−0.691369 + 0.722502i \(0.742992\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 27.0000 1.27421 0.637104 0.770778i \(-0.280132\pi\)
0.637104 + 0.770778i \(0.280132\pi\)
\(450\) 0 0
\(451\) −13.2288 −0.622918
\(452\) 0 0
\(453\) 10.5830 0.497233
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 27.0000 1.26301 0.631503 0.775373i \(-0.282438\pi\)
0.631503 + 0.775373i \(0.282438\pi\)
\(458\) 0 0
\(459\) 7.93725 0.370479
\(460\) 0 0
\(461\) −42.3320 −1.97160 −0.985799 0.167927i \(-0.946293\pi\)
−0.985799 + 0.167927i \(0.946293\pi\)
\(462\) 0 0
\(463\) −8.00000 −0.371792 −0.185896 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −26.4575 −1.22431 −0.612154 0.790739i \(-0.709697\pi\)
−0.612154 + 0.790739i \(0.709697\pi\)
\(468\) 0 0
\(469\) −31.7490 −1.46603
\(470\) 0 0
\(471\) −28.0000 −1.29017
\(472\) 0 0
\(473\) −14.0000 −0.643721
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −42.3320 −1.93825
\(478\) 0 0
\(479\) 4.00000 0.182765 0.0913823 0.995816i \(-0.470871\pi\)
0.0913823 + 0.995816i \(0.470871\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) −42.3320 −1.92617
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 12.0000 0.543772 0.271886 0.962329i \(-0.412353\pi\)
0.271886 + 0.962329i \(0.412353\pi\)
\(488\) 0 0
\(489\) 35.0000 1.58275
\(490\) 0 0
\(491\) −5.29150 −0.238802 −0.119401 0.992846i \(-0.538097\pi\)
−0.119401 + 0.992846i \(0.538097\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 32.0000 1.43540
\(498\) 0 0
\(499\) 26.4575 1.18440 0.592200 0.805791i \(-0.298259\pi\)
0.592200 + 0.805791i \(0.298259\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 34.3948 1.52753
\(508\) 0 0
\(509\) −31.7490 −1.40725 −0.703625 0.710571i \(-0.748437\pi\)
−0.703625 + 0.710571i \(0.748437\pi\)
\(510\) 0 0
\(511\) −28.0000 −1.23865
\(512\) 0 0
\(513\) −7.00000 −0.309058
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 21.1660 0.930880
\(518\) 0 0
\(519\) 56.0000 2.45813
\(520\) 0 0
\(521\) −3.00000 −0.131432 −0.0657162 0.997838i \(-0.520933\pi\)
−0.0657162 + 0.997838i \(0.520933\pi\)
\(522\) 0 0
\(523\) 2.64575 0.115691 0.0578453 0.998326i \(-0.481577\pi\)
0.0578453 + 0.998326i \(0.481577\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −12.0000 −0.522728
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −21.1660 −0.918527
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −63.0000 −2.71865
\(538\) 0 0
\(539\) −23.8118 −1.02565
\(540\) 0 0
\(541\) −21.1660 −0.909998 −0.454999 0.890492i \(-0.650360\pi\)
−0.454999 + 0.890492i \(0.650360\pi\)
\(542\) 0 0
\(543\) 28.0000 1.20160
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 18.5203 0.791869 0.395935 0.918279i \(-0.370421\pi\)
0.395935 + 0.918279i \(0.370421\pi\)
\(548\) 0 0
\(549\) 42.3320 1.80669
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −16.0000 −0.680389
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −21.0000 −0.886621
\(562\) 0 0
\(563\) 15.8745 0.669031 0.334515 0.942390i \(-0.391427\pi\)
0.334515 + 0.942390i \(0.391427\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 20.0000 0.839921
\(568\) 0 0
\(569\) −11.0000 −0.461144 −0.230572 0.973055i \(-0.574060\pi\)
−0.230572 + 0.973055i \(0.574060\pi\)
\(570\) 0 0
\(571\) 37.0405 1.55010 0.775049 0.631901i \(-0.217725\pi\)
0.775049 + 0.631901i \(0.217725\pi\)
\(572\) 0 0
\(573\) −10.5830 −0.442111
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −7.00000 −0.291414 −0.145707 0.989328i \(-0.546546\pi\)
−0.145707 + 0.989328i \(0.546546\pi\)
\(578\) 0 0
\(579\) −13.2288 −0.549768
\(580\) 0 0
\(581\) −31.7490 −1.31717
\(582\) 0 0
\(583\) 28.0000 1.15964
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −7.93725 −0.327606 −0.163803 0.986493i \(-0.552376\pi\)
−0.163803 + 0.986493i \(0.552376\pi\)
\(588\) 0 0
\(589\) 10.5830 0.436065
\(590\) 0 0
\(591\) 28.0000 1.15177
\(592\) 0 0
\(593\) 41.0000 1.68367 0.841834 0.539736i \(-0.181476\pi\)
0.841834 + 0.539736i \(0.181476\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −63.4980 −2.59880
\(598\) 0 0
\(599\) 12.0000 0.490307 0.245153 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) 0 0
\(601\) −7.00000 −0.285536 −0.142768 0.989756i \(-0.545600\pi\)
−0.142768 + 0.989756i \(0.545600\pi\)
\(602\) 0 0
\(603\) 31.7490 1.29292
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −8.00000 −0.324710 −0.162355 0.986732i \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 10.5830 0.427444 0.213722 0.976895i \(-0.431441\pi\)
0.213722 + 0.976895i \(0.431441\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 0 0
\(619\) −5.29150 −0.212683 −0.106342 0.994330i \(-0.533914\pi\)
−0.106342 + 0.994330i \(0.533914\pi\)
\(620\) 0 0
\(621\) 10.5830 0.424681
\(622\) 0 0
\(623\) −4.00000 −0.160257
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 18.5203 0.739628
\(628\) 0 0
\(629\) −31.7490 −1.26592
\(630\) 0 0
\(631\) −36.0000 −1.43314 −0.716569 0.697517i \(-0.754288\pi\)
−0.716569 + 0.697517i \(0.754288\pi\)
\(632\) 0 0
\(633\) 21.0000 0.834675
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −32.0000 −1.26590
\(640\) 0 0
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) 0 0
\(643\) 15.8745 0.626029 0.313015 0.949748i \(-0.398661\pi\)
0.313015 + 0.949748i \(0.398661\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 14.0000 0.549548
\(650\) 0 0
\(651\) 42.3320 1.65912
\(652\) 0 0
\(653\) −31.7490 −1.24243 −0.621217 0.783638i \(-0.713362\pi\)
−0.621217 + 0.783638i \(0.713362\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 28.0000 1.09238
\(658\) 0 0
\(659\) −7.93725 −0.309192 −0.154596 0.987978i \(-0.549408\pi\)
−0.154596 + 0.987978i \(0.549408\pi\)
\(660\) 0 0
\(661\) −21.1660 −0.823262 −0.411631 0.911351i \(-0.635041\pi\)
−0.411631 + 0.911351i \(0.635041\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 42.3320 1.63665
\(670\) 0 0
\(671\) −28.0000 −1.08093
\(672\) 0 0
\(673\) −34.0000 −1.31060 −0.655302 0.755367i \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 42.3320 1.62695 0.813476 0.581599i \(-0.197573\pi\)
0.813476 + 0.581599i \(0.197573\pi\)
\(678\) 0 0
\(679\) 8.00000 0.307012
\(680\) 0 0
\(681\) −42.0000 −1.60944
\(682\) 0 0
\(683\) 23.8118 0.911132 0.455566 0.890202i \(-0.349437\pi\)
0.455566 + 0.890202i \(0.349437\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 56.0000 2.13653
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 23.8118 0.905842 0.452921 0.891551i \(-0.350382\pi\)
0.452921 + 0.891551i \(0.350382\pi\)
\(692\) 0 0
\(693\) 42.3320 1.60806
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −15.0000 −0.568166
\(698\) 0 0
\(699\) 15.8745 0.600429
\(700\) 0 0
\(701\) 21.1660 0.799429 0.399715 0.916640i \(-0.369109\pi\)
0.399715 + 0.916640i \(0.369109\pi\)
\(702\) 0 0
\(703\) 28.0000 1.05604
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −42.3320 −1.59206
\(708\) 0 0
\(709\) −21.1660 −0.794906 −0.397453 0.917622i \(-0.630106\pi\)
−0.397453 + 0.917622i \(0.630106\pi\)
\(710\) 0 0
\(711\) 16.0000 0.600047
\(712\) 0 0
\(713\) −16.0000 −0.599205
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 21.1660 0.790459
\(718\) 0 0
\(719\) 36.0000 1.34257 0.671287 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) 0 0
\(721\) 32.0000 1.19174
\(722\) 0 0
\(723\) 55.5608 2.06633
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 16.0000 0.593407 0.296704 0.954970i \(-0.404113\pi\)
0.296704 + 0.954970i \(0.404113\pi\)
\(728\) 0 0
\(729\) −41.0000 −1.51852
\(730\) 0 0
\(731\) −15.8745 −0.587140
\(732\) 0 0
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −21.0000 −0.773545
\(738\) 0 0
\(739\) −15.8745 −0.583953 −0.291977 0.956425i \(-0.594313\pi\)
−0.291977 + 0.956425i \(0.594313\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −36.0000 −1.32071 −0.660356 0.750953i \(-0.729595\pi\)
−0.660356 + 0.750953i \(0.729595\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 31.7490 1.16164
\(748\) 0 0
\(749\) −10.5830 −0.386695
\(750\) 0 0
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) 0 0
\(753\) −21.0000 −0.765283
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −21.1660 −0.769292 −0.384646 0.923064i \(-0.625676\pi\)
−0.384646 + 0.923064i \(0.625676\pi\)
\(758\) 0 0
\(759\) −28.0000 −1.01634
\(760\) 0 0
\(761\) 29.0000 1.05125 0.525625 0.850717i \(-0.323832\pi\)
0.525625 + 0.850717i \(0.323832\pi\)
\(762\) 0 0
\(763\) −42.3320 −1.53252
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −21.0000 −0.757279 −0.378640 0.925544i \(-0.623608\pi\)
−0.378640 + 0.925544i \(0.623608\pi\)
\(770\) 0 0
\(771\) −37.0405 −1.33398
\(772\) 0 0
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 112.000 4.01798
\(778\) 0 0
\(779\) 13.2288 0.473969
\(780\) 0 0
\(781\) 21.1660 0.757379
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −26.4575 −0.943108 −0.471554 0.881837i \(-0.656307\pi\)
−0.471554 + 0.881837i \(0.656307\pi\)
\(788\) 0 0
\(789\) 31.7490 1.13029
\(790\) 0 0
\(791\) 60.0000 2.13335
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 31.7490 1.12461 0.562304 0.826931i \(-0.309915\pi\)
0.562304 + 0.826931i \(0.309915\pi\)
\(798\) 0 0
\(799\) 24.0000 0.849059
\(800\) 0 0
\(801\) 4.00000 0.141333
\(802\) 0 0
\(803\) −18.5203 −0.653566
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 56.0000 1.97129
\(808\) 0 0
\(809\) −10.0000 −0.351581 −0.175791 0.984428i \(-0.556248\pi\)
−0.175791 + 0.984428i \(0.556248\pi\)
\(810\) 0 0
\(811\) −5.29150 −0.185810 −0.0929049 0.995675i \(-0.529615\pi\)
−0.0929049 + 0.995675i \(0.529615\pi\)
\(812\) 0 0
\(813\) 52.9150 1.85581
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 14.0000 0.489798
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 10.5830 0.369349 0.184675 0.982800i \(-0.440877\pi\)
0.184675 + 0.982800i \(0.440877\pi\)
\(822\) 0 0
\(823\) 24.0000 0.836587 0.418294 0.908312i \(-0.362628\pi\)
0.418294 + 0.908312i \(0.362628\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 13.2288 0.460009 0.230004 0.973190i \(-0.426126\pi\)
0.230004 + 0.973190i \(0.426126\pi\)
\(828\) 0 0
\(829\) −21.1660 −0.735126 −0.367563 0.929999i \(-0.619808\pi\)
−0.367563 + 0.929999i \(0.619808\pi\)
\(830\) 0 0
\(831\) −56.0000 −1.94262
\(832\) 0 0
\(833\) −27.0000 −0.935495
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −10.5830 −0.365802
\(838\) 0 0
\(839\) −40.0000 −1.38095 −0.690477 0.723355i \(-0.742599\pi\)
−0.690477 + 0.723355i \(0.742599\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) −58.2065 −2.00474
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 16.0000 0.549767
\(848\) 0 0
\(849\) −35.0000 −1.20120
\(850\) 0 0
\(851\) −42.3320 −1.45112
\(852\) 0 0
\(853\) −52.9150 −1.81178 −0.905888 0.423517i \(-0.860795\pi\)
−0.905888 + 0.423517i \(0.860795\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −21.0000 −0.717346 −0.358673 0.933463i \(-0.616771\pi\)
−0.358673 + 0.933463i \(0.616771\pi\)
\(858\) 0 0
\(859\) −2.64575 −0.0902719 −0.0451359 0.998981i \(-0.514372\pi\)
−0.0451359 + 0.998981i \(0.514372\pi\)
\(860\) 0 0
\(861\) 52.9150 1.80334
\(862\) 0 0
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 21.1660 0.718835
\(868\) 0 0
\(869\) −10.5830 −0.359004
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −8.00000 −0.270759
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 42.3320 1.42945 0.714725 0.699405i \(-0.246552\pi\)
0.714725 + 0.699405i \(0.246552\pi\)
\(878\) 0 0
\(879\) −28.0000 −0.944417
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 0 0
\(883\) −44.9778 −1.51362 −0.756811 0.653633i \(-0.773244\pi\)
−0.756811 + 0.653633i \(0.773244\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −56.0000 −1.88030 −0.940148 0.340766i \(-0.889313\pi\)
−0.940148 + 0.340766i \(0.889313\pi\)
\(888\) 0 0
\(889\) −48.0000 −1.60987
\(890\) 0 0
\(891\) 13.2288 0.443180
\(892\) 0 0
\(893\) −21.1660 −0.708294
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 31.7490 1.05771
\(902\) 0 0
\(903\) 56.0000 1.86356
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 5.29150 0.175701 0.0878507 0.996134i \(-0.472000\pi\)
0.0878507 + 0.996134i \(0.472000\pi\)
\(908\) 0 0
\(909\) 42.3320 1.40406
\(910\) 0 0
\(911\) 48.0000 1.59031 0.795155 0.606406i \(-0.207389\pi\)
0.795155 + 0.606406i \(0.207389\pi\)
\(912\) 0 0
\(913\) −21.0000 −0.694999
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 63.4980 2.09689
\(918\) 0 0
\(919\) −4.00000 −0.131948 −0.0659739 0.997821i \(-0.521015\pi\)
−0.0659739 + 0.997821i \(0.521015\pi\)
\(920\) 0 0
\(921\) 7.00000 0.230658
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −32.0000 −1.05102
\(928\) 0 0
\(929\) −14.0000 −0.459325 −0.229663 0.973270i \(-0.573762\pi\)
−0.229663 + 0.973270i \(0.573762\pi\)
\(930\) 0 0
\(931\) 23.8118 0.780399
\(932\) 0 0
\(933\) −10.5830 −0.346472
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 7.00000 0.228680 0.114340 0.993442i \(-0.463525\pi\)
0.114340 + 0.993442i \(0.463525\pi\)
\(938\) 0 0
\(939\) 15.8745 0.518045
\(940\) 0 0
\(941\) 42.3320 1.37998 0.689992 0.723817i \(-0.257614\pi\)
0.689992 + 0.723817i \(0.257614\pi\)
\(942\) 0 0
\(943\) −20.0000 −0.651290
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 15.8745 0.515852 0.257926 0.966165i \(-0.416961\pi\)
0.257926 + 0.966165i \(0.416961\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −5.00000 −0.161966 −0.0809829 0.996715i \(-0.525806\pi\)
−0.0809829 + 0.996715i \(0.525806\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −76.0000 −2.45417
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 10.5830 0.341033
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −8.00000 −0.257263 −0.128631 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) 0 0
\(969\) 21.0000 0.674617
\(970\) 0 0
\(971\) −23.8118 −0.764156 −0.382078 0.924130i \(-0.624791\pi\)
−0.382078 + 0.924130i \(0.624791\pi\)
\(972\) 0 0
\(973\) −74.0810 −2.37493
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 37.0000 1.18373 0.591867 0.806035i \(-0.298391\pi\)
0.591867 + 0.806035i \(0.298391\pi\)
\(978\) 0 0
\(979\) −2.64575 −0.0845586
\(980\) 0 0
\(981\) 42.3320 1.35156
\(982\) 0 0
\(983\) 28.0000 0.893061 0.446531 0.894768i \(-0.352659\pi\)
0.446531 + 0.894768i \(0.352659\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −84.6640 −2.69489
\(988\) 0 0
\(989\) −21.1660 −0.673040
\(990\) 0 0
\(991\) 44.0000 1.39771 0.698853 0.715265i \(-0.253694\pi\)
0.698853 + 0.715265i \(0.253694\pi\)
\(992\) 0 0
\(993\) 7.00000 0.222138
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −42.3320 −1.34067 −0.670334 0.742059i \(-0.733849\pi\)
−0.670334 + 0.742059i \(0.733849\pi\)
\(998\) 0 0
\(999\) −28.0000 −0.885881
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 6400.2.a.bg.1.1 2
4.3 odd 2 6400.2.a.cb.1.2 2
5.4 even 2 6400.2.a.cc.1.2 2
8.3 odd 2 6400.2.a.cb.1.1 2
8.5 even 2 inner 6400.2.a.bg.1.2 2
16.3 odd 4 800.2.d.a.401.2 2
16.5 even 4 200.2.d.c.101.2 yes 2
16.11 odd 4 800.2.d.a.401.1 2
16.13 even 4 200.2.d.c.101.1 yes 2
20.19 odd 2 6400.2.a.bh.1.1 2
40.19 odd 2 6400.2.a.bh.1.2 2
40.29 even 2 6400.2.a.cc.1.1 2
48.5 odd 4 1800.2.k.d.901.1 2
48.11 even 4 7200.2.k.b.3601.1 2
48.29 odd 4 1800.2.k.d.901.2 2
48.35 even 4 7200.2.k.b.3601.2 2
80.3 even 4 800.2.f.d.49.2 4
80.13 odd 4 200.2.f.d.149.1 4
80.19 odd 4 800.2.d.d.401.1 2
80.27 even 4 800.2.f.d.49.1 4
80.29 even 4 200.2.d.b.101.2 yes 2
80.37 odd 4 200.2.f.d.149.2 4
80.43 even 4 800.2.f.d.49.4 4
80.53 odd 4 200.2.f.d.149.3 4
80.59 odd 4 800.2.d.d.401.2 2
80.67 even 4 800.2.f.d.49.3 4
80.69 even 4 200.2.d.b.101.1 2
80.77 odd 4 200.2.f.d.149.4 4
240.29 odd 4 1800.2.k.f.901.1 2
240.53 even 4 1800.2.d.m.1549.2 4
240.59 even 4 7200.2.k.i.3601.1 2
240.77 even 4 1800.2.d.m.1549.1 4
240.83 odd 4 7200.2.d.m.2449.4 4
240.107 odd 4 7200.2.d.m.2449.1 4
240.149 odd 4 1800.2.k.f.901.2 2
240.173 even 4 1800.2.d.m.1549.4 4
240.179 even 4 7200.2.k.i.3601.2 2
240.197 even 4 1800.2.d.m.1549.3 4
240.203 odd 4 7200.2.d.m.2449.3 4
240.227 odd 4 7200.2.d.m.2449.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
200.2.d.b.101.1 2 80.69 even 4
200.2.d.b.101.2 yes 2 80.29 even 4
200.2.d.c.101.1 yes 2 16.13 even 4
200.2.d.c.101.2 yes 2 16.5 even 4
200.2.f.d.149.1 4 80.13 odd 4
200.2.f.d.149.2 4 80.37 odd 4
200.2.f.d.149.3 4 80.53 odd 4
200.2.f.d.149.4 4 80.77 odd 4
800.2.d.a.401.1 2 16.11 odd 4
800.2.d.a.401.2 2 16.3 odd 4
800.2.d.d.401.1 2 80.19 odd 4
800.2.d.d.401.2 2 80.59 odd 4
800.2.f.d.49.1 4 80.27 even 4
800.2.f.d.49.2 4 80.3 even 4
800.2.f.d.49.3 4 80.67 even 4
800.2.f.d.49.4 4 80.43 even 4
1800.2.d.m.1549.1 4 240.77 even 4
1800.2.d.m.1549.2 4 240.53 even 4
1800.2.d.m.1549.3 4 240.197 even 4
1800.2.d.m.1549.4 4 240.173 even 4
1800.2.k.d.901.1 2 48.5 odd 4
1800.2.k.d.901.2 2 48.29 odd 4
1800.2.k.f.901.1 2 240.29 odd 4
1800.2.k.f.901.2 2 240.149 odd 4
6400.2.a.bg.1.1 2 1.1 even 1 trivial
6400.2.a.bg.1.2 2 8.5 even 2 inner
6400.2.a.bh.1.1 2 20.19 odd 2
6400.2.a.bh.1.2 2 40.19 odd 2
6400.2.a.cb.1.1 2 8.3 odd 2
6400.2.a.cb.1.2 2 4.3 odd 2
6400.2.a.cc.1.1 2 40.29 even 2
6400.2.a.cc.1.2 2 5.4 even 2
7200.2.d.m.2449.1 4 240.107 odd 4
7200.2.d.m.2449.2 4 240.227 odd 4
7200.2.d.m.2449.3 4 240.203 odd 4
7200.2.d.m.2449.4 4 240.83 odd 4
7200.2.k.b.3601.1 2 48.11 even 4
7200.2.k.b.3601.2 2 48.35 even 4
7200.2.k.i.3601.1 2 240.59 even 4
7200.2.k.i.3601.2 2 240.179 even 4