Properties

Label 7800.2.a.bf.1.3
Level $7800$
Weight $2$
Character 7800.1
Self dual yes
Analytic conductor $62.283$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(62.2833135766\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.1849.1
Defining polynomial: \(x^{3} - x^{2} - 14 x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1560)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(4.50331\) of defining polynomial
Character \(\chi\) \(=\) 7800.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +2.50331 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +2.50331 q^{7} +1.00000 q^{9} +3.27316 q^{11} -1.00000 q^{13} -4.50331 q^{17} +3.23014 q^{19} -2.50331 q^{21} +2.50331 q^{23} -1.00000 q^{27} -7.77647 q^{29} -5.00662 q^{31} -3.27316 q^{33} -7.73345 q^{37} +1.00000 q^{39} +7.73345 q^{41} -4.00000 q^{43} -9.00662 q^{47} -0.733451 q^{49} +4.50331 q^{51} -6.27978 q^{53} -3.23014 q^{57} +9.27316 q^{61} +2.50331 q^{63} -15.5529 q^{67} -2.50331 q^{69} +5.73345 q^{71} -15.7765 q^{73} +8.19374 q^{77} -4.72684 q^{79} +1.00000 q^{81} +11.5529 q^{83} +7.77647 q^{87} -15.2864 q^{89} -2.50331 q^{91} +5.00662 q^{93} -12.5033 q^{97} +3.27316 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{3} - 5q^{7} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{3} - 5q^{7} + 3q^{9} + 3q^{11} - 3q^{13} - q^{17} + 4q^{19} + 5q^{21} - 5q^{23} - 3q^{27} - 4q^{29} + 10q^{31} - 3q^{33} - 5q^{37} + 3q^{39} + 5q^{41} - 12q^{43} - 2q^{47} + 16q^{49} + q^{51} + 13q^{53} - 4q^{57} + 21q^{61} - 5q^{63} - 8q^{67} + 5q^{69} - q^{71} - 28q^{73} - 5q^{77} - 21q^{79} + 3q^{81} - 4q^{83} + 4q^{87} + 11q^{89} + 5q^{91} - 10q^{93} - 25q^{97} + 3q^{99} + O(q^{100}) \)

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
\(7\) 2.50331 0.946161 0.473081 0.881019i \(-0.343142\pi\)
0.473081 + 0.881019i \(0.343142\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.27316 0.986896 0.493448 0.869775i \(-0.335736\pi\)
0.493448 + 0.869775i \(0.335736\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.50331 −1.09221 −0.546106 0.837716i \(-0.683891\pi\)
−0.546106 + 0.837716i \(0.683891\pi\)
\(18\) 0 0
\(19\) 3.23014 0.741046 0.370523 0.928823i \(-0.379179\pi\)
0.370523 + 0.928823i \(0.379179\pi\)
\(20\) 0 0
\(21\) −2.50331 −0.546267
\(22\) 0 0
\(23\) 2.50331 0.521976 0.260988 0.965342i \(-0.415952\pi\)
0.260988 + 0.965342i \(0.415952\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −7.77647 −1.44405 −0.722027 0.691865i \(-0.756790\pi\)
−0.722027 + 0.691865i \(0.756790\pi\)
\(30\) 0 0
\(31\) −5.00662 −0.899215 −0.449607 0.893226i \(-0.648436\pi\)
−0.449607 + 0.893226i \(0.648436\pi\)
\(32\) 0 0
\(33\) −3.27316 −0.569785
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −7.73345 −1.27137 −0.635686 0.771948i \(-0.719283\pi\)
−0.635686 + 0.771948i \(0.719283\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 7.73345 1.20776 0.603881 0.797074i \(-0.293620\pi\)
0.603881 + 0.797074i \(0.293620\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −9.00662 −1.31375 −0.656875 0.754000i \(-0.728122\pi\)
−0.656875 + 0.754000i \(0.728122\pi\)
\(48\) 0 0
\(49\) −0.733451 −0.104779
\(50\) 0 0
\(51\) 4.50331 0.630589
\(52\) 0 0
\(53\) −6.27978 −0.862594 −0.431297 0.902210i \(-0.641944\pi\)
−0.431297 + 0.902210i \(0.641944\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −3.23014 −0.427843
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 9.27316 1.18731 0.593654 0.804721i \(-0.297685\pi\)
0.593654 + 0.804721i \(0.297685\pi\)
\(62\) 0 0
\(63\) 2.50331 0.315387
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −15.5529 −1.90009 −0.950047 0.312106i \(-0.898966\pi\)
−0.950047 + 0.312106i \(0.898966\pi\)
\(68\) 0 0
\(69\) −2.50331 −0.301363
\(70\) 0 0
\(71\) 5.73345 0.680435 0.340218 0.940347i \(-0.389499\pi\)
0.340218 + 0.940347i \(0.389499\pi\)
\(72\) 0 0
\(73\) −15.7765 −1.84650 −0.923248 0.384204i \(-0.874476\pi\)
−0.923248 + 0.384204i \(0.874476\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 8.19374 0.933763
\(78\) 0 0
\(79\) −4.72684 −0.531811 −0.265905 0.963999i \(-0.585671\pi\)
−0.265905 + 0.963999i \(0.585671\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 11.5529 1.26810 0.634050 0.773292i \(-0.281391\pi\)
0.634050 + 0.773292i \(0.281391\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 7.77647 0.833725
\(88\) 0 0
\(89\) −15.2864 −1.62035 −0.810177 0.586185i \(-0.800629\pi\)
−0.810177 + 0.586185i \(0.800629\pi\)
\(90\) 0 0
\(91\) −2.50331 −0.262418
\(92\) 0 0
\(93\) 5.00662 0.519162
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −12.5033 −1.26952 −0.634759 0.772710i \(-0.718901\pi\)
−0.634759 + 0.772710i \(0.718901\pi\)
\(98\) 0 0
\(99\) 3.27316 0.328965
\(100\) 0 0
\(101\) 5.23014 0.520419 0.260209 0.965552i \(-0.416208\pi\)
0.260209 + 0.965552i \(0.416208\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0.726836 0.0702659 0.0351329 0.999383i \(-0.488815\pi\)
0.0351329 + 0.999383i \(0.488815\pi\)
\(108\) 0 0
\(109\) 16.7831 1.60753 0.803764 0.594948i \(-0.202827\pi\)
0.803764 + 0.594948i \(0.202827\pi\)
\(110\) 0 0
\(111\) 7.73345 0.734027
\(112\) 0 0
\(113\) 3.77647 0.355261 0.177630 0.984097i \(-0.443157\pi\)
0.177630 + 0.984097i \(0.443157\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) −11.2732 −1.03341
\(120\) 0 0
\(121\) −0.286395 −0.0260359
\(122\) 0 0
\(123\) −7.73345 −0.697302
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 6.54633 0.580893 0.290446 0.956891i \(-0.406196\pi\)
0.290446 + 0.956891i \(0.406196\pi\)
\(128\) 0 0
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) −11.2301 −0.981182 −0.490591 0.871390i \(-0.663219\pi\)
−0.490591 + 0.871390i \(0.663219\pi\)
\(132\) 0 0
\(133\) 8.08604 0.701149
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 13.5529 1.15791 0.578953 0.815361i \(-0.303461\pi\)
0.578953 + 0.815361i \(0.303461\pi\)
\(138\) 0 0
\(139\) −21.7335 −1.84341 −0.921704 0.387895i \(-0.873202\pi\)
−0.921704 + 0.387895i \(0.873202\pi\)
\(140\) 0 0
\(141\) 9.00662 0.758494
\(142\) 0 0
\(143\) −3.27316 −0.273716
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0.733451 0.0604940
\(148\) 0 0
\(149\) 2.27978 0.186767 0.0933834 0.995630i \(-0.470232\pi\)
0.0933834 + 0.995630i \(0.470232\pi\)
\(150\) 0 0
\(151\) −6.46029 −0.525731 −0.262865 0.964833i \(-0.584667\pi\)
−0.262865 + 0.964833i \(0.584667\pi\)
\(152\) 0 0
\(153\) −4.50331 −0.364071
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 3.53971 0.282500 0.141250 0.989974i \(-0.454888\pi\)
0.141250 + 0.989974i \(0.454888\pi\)
\(158\) 0 0
\(159\) 6.27978 0.498019
\(160\) 0 0
\(161\) 6.26655 0.493873
\(162\) 0 0
\(163\) 18.7401 1.46784 0.733918 0.679238i \(-0.237690\pi\)
0.733918 + 0.679238i \(0.237690\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 4.00000 0.309529 0.154765 0.987951i \(-0.450538\pi\)
0.154765 + 0.987951i \(0.450538\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 3.23014 0.247015
\(172\) 0 0
\(173\) 8.46029 0.643224 0.321612 0.946872i \(-0.395775\pi\)
0.321612 + 0.946872i \(0.395775\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −4.76986 −0.356516 −0.178258 0.983984i \(-0.557046\pi\)
−0.178258 + 0.983984i \(0.557046\pi\)
\(180\) 0 0
\(181\) 15.7335 1.16946 0.584729 0.811229i \(-0.301201\pi\)
0.584729 + 0.811229i \(0.301201\pi\)
\(182\) 0 0
\(183\) −9.27316 −0.685492
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −14.7401 −1.07790
\(188\) 0 0
\(189\) −2.50331 −0.182089
\(190\) 0 0
\(191\) −26.0132 −1.88225 −0.941126 0.338057i \(-0.890230\pi\)
−0.941126 + 0.338057i \(0.890230\pi\)
\(192\) 0 0
\(193\) 0.503308 0.0362289 0.0181144 0.999836i \(-0.494234\pi\)
0.0181144 + 0.999836i \(0.494234\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 11.0066 0.784189 0.392094 0.919925i \(-0.371751\pi\)
0.392094 + 0.919925i \(0.371751\pi\)
\(198\) 0 0
\(199\) 2.01323 0.142714 0.0713571 0.997451i \(-0.477267\pi\)
0.0713571 + 0.997451i \(0.477267\pi\)
\(200\) 0 0
\(201\) 15.5529 1.09702
\(202\) 0 0
\(203\) −19.4669 −1.36631
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 2.50331 0.173992
\(208\) 0 0
\(209\) 10.5728 0.731335
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 0 0
\(213\) −5.73345 −0.392850
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −12.5331 −0.850802
\(218\) 0 0
\(219\) 15.7765 1.06608
\(220\) 0 0
\(221\) 4.50331 0.302925
\(222\) 0 0
\(223\) −23.7897 −1.59308 −0.796538 0.604588i \(-0.793338\pi\)
−0.796538 + 0.604588i \(0.793338\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 23.0198 1.52788 0.763940 0.645287i \(-0.223262\pi\)
0.763940 + 0.645287i \(0.223262\pi\)
\(228\) 0 0
\(229\) −7.77647 −0.513884 −0.256942 0.966427i \(-0.582715\pi\)
−0.256942 + 0.966427i \(0.582715\pi\)
\(230\) 0 0
\(231\) −8.19374 −0.539108
\(232\) 0 0
\(233\) −23.9702 −1.57034 −0.785170 0.619280i \(-0.787425\pi\)
−0.785170 + 0.619280i \(0.787425\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 4.72684 0.307041
\(238\) 0 0
\(239\) 0.812878 0.0525807 0.0262903 0.999654i \(-0.491631\pi\)
0.0262903 + 0.999654i \(0.491631\pi\)
\(240\) 0 0
\(241\) 20.0132 1.28917 0.644583 0.764535i \(-0.277031\pi\)
0.644583 + 0.764535i \(0.277031\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −3.23014 −0.205529
\(248\) 0 0
\(249\) −11.5529 −0.732138
\(250\) 0 0
\(251\) −11.3162 −0.714271 −0.357136 0.934053i \(-0.616247\pi\)
−0.357136 + 0.934053i \(0.616247\pi\)
\(252\) 0 0
\(253\) 8.19374 0.515136
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −19.3294 −1.20574 −0.602868 0.797841i \(-0.705975\pi\)
−0.602868 + 0.797841i \(0.705975\pi\)
\(258\) 0 0
\(259\) −19.3592 −1.20292
\(260\) 0 0
\(261\) −7.77647 −0.481352
\(262\) 0 0
\(263\) 12.2368 0.754551 0.377275 0.926101i \(-0.376861\pi\)
0.377275 + 0.926101i \(0.376861\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 15.2864 0.935512
\(268\) 0 0
\(269\) −2.76986 −0.168881 −0.0844406 0.996429i \(-0.526910\pi\)
−0.0844406 + 0.996429i \(0.526910\pi\)
\(270\) 0 0
\(271\) −0.0860423 −0.00522670 −0.00261335 0.999997i \(-0.500832\pi\)
−0.00261335 + 0.999997i \(0.500832\pi\)
\(272\) 0 0
\(273\) 2.50331 0.151507
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −17.5529 −1.05465 −0.527327 0.849662i \(-0.676806\pi\)
−0.527327 + 0.849662i \(0.676806\pi\)
\(278\) 0 0
\(279\) −5.00662 −0.299738
\(280\) 0 0
\(281\) 24.0132 1.43251 0.716255 0.697839i \(-0.245855\pi\)
0.716255 + 0.697839i \(0.245855\pi\)
\(282\) 0 0
\(283\) −22.0132 −1.30855 −0.654275 0.756256i \(-0.727026\pi\)
−0.654275 + 0.756256i \(0.727026\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 19.3592 1.14274
\(288\) 0 0
\(289\) 3.27978 0.192928
\(290\) 0 0
\(291\) 12.5033 0.732957
\(292\) 0 0
\(293\) 8.99338 0.525399 0.262700 0.964878i \(-0.415387\pi\)
0.262700 + 0.964878i \(0.415387\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −3.27316 −0.189928
\(298\) 0 0
\(299\) −2.50331 −0.144770
\(300\) 0 0
\(301\) −10.0132 −0.577153
\(302\) 0 0
\(303\) −5.23014 −0.300464
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 22.2930 1.27233 0.636165 0.771553i \(-0.280520\pi\)
0.636165 + 0.771553i \(0.280520\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 17.5529 0.992151 0.496076 0.868279i \(-0.334774\pi\)
0.496076 + 0.868279i \(0.334774\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 4.54633 0.255347 0.127674 0.991816i \(-0.459249\pi\)
0.127674 + 0.991816i \(0.459249\pi\)
\(318\) 0 0
\(319\) −25.4537 −1.42513
\(320\) 0 0
\(321\) −0.726836 −0.0405680
\(322\) 0 0
\(323\) −14.5463 −0.809379
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −16.7831 −0.928107
\(328\) 0 0
\(329\) −22.5463 −1.24302
\(330\) 0 0
\(331\) 19.7897 1.08774 0.543870 0.839169i \(-0.316958\pi\)
0.543870 + 0.839169i \(0.316958\pi\)
\(332\) 0 0
\(333\) −7.73345 −0.423790
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 13.0198 0.709236 0.354618 0.935011i \(-0.384611\pi\)
0.354618 + 0.935011i \(0.384611\pi\)
\(338\) 0 0
\(339\) −3.77647 −0.205110
\(340\) 0 0
\(341\) −16.3875 −0.887432
\(342\) 0 0
\(343\) −19.3592 −1.04530
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 7.18712 0.385825 0.192912 0.981216i \(-0.438207\pi\)
0.192912 + 0.981216i \(0.438207\pi\)
\(348\) 0 0
\(349\) −17.2301 −0.922308 −0.461154 0.887320i \(-0.652565\pi\)
−0.461154 + 0.887320i \(0.652565\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) 31.5662 1.68010 0.840049 0.542511i \(-0.182526\pi\)
0.840049 + 0.542511i \(0.182526\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 11.2732 0.596639
\(358\) 0 0
\(359\) −2.46029 −0.129849 −0.0649245 0.997890i \(-0.520681\pi\)
−0.0649245 + 0.997890i \(0.520681\pi\)
\(360\) 0 0
\(361\) −8.56617 −0.450851
\(362\) 0 0
\(363\) 0.286395 0.0150318
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −5.09266 −0.265835 −0.132917 0.991127i \(-0.542434\pi\)
−0.132917 + 0.991127i \(0.542434\pi\)
\(368\) 0 0
\(369\) 7.73345 0.402587
\(370\) 0 0
\(371\) −15.7202 −0.816153
\(372\) 0 0
\(373\) −11.0066 −0.569901 −0.284950 0.958542i \(-0.591977\pi\)
−0.284950 + 0.958542i \(0.591977\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 7.77647 0.400509
\(378\) 0 0
\(379\) −26.3360 −1.35279 −0.676396 0.736539i \(-0.736459\pi\)
−0.676396 + 0.736539i \(0.736459\pi\)
\(380\) 0 0
\(381\) −6.54633 −0.335379
\(382\) 0 0
\(383\) −18.5463 −0.947673 −0.473837 0.880613i \(-0.657131\pi\)
−0.473837 + 0.880613i \(0.657131\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −4.00000 −0.203331
\(388\) 0 0
\(389\) −18.7699 −0.951670 −0.475835 0.879535i \(-0.657854\pi\)
−0.475835 + 0.879535i \(0.657854\pi\)
\(390\) 0 0
\(391\) −11.2732 −0.570108
\(392\) 0 0
\(393\) 11.2301 0.566486
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0.266549 0.0133777 0.00668886 0.999978i \(-0.497871\pi\)
0.00668886 + 0.999978i \(0.497871\pi\)
\(398\) 0 0
\(399\) −8.08604 −0.404808
\(400\) 0 0
\(401\) 35.5662 1.77609 0.888045 0.459757i \(-0.152063\pi\)
0.888045 + 0.459757i \(0.152063\pi\)
\(402\) 0 0
\(403\) 5.00662 0.249397
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −25.3129 −1.25471
\(408\) 0 0
\(409\) 0.460287 0.0227597 0.0113799 0.999935i \(-0.496378\pi\)
0.0113799 + 0.999935i \(0.496378\pi\)
\(410\) 0 0
\(411\) −13.5529 −0.668517
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 21.7335 1.06429
\(418\) 0 0
\(419\) −8.32280 −0.406595 −0.203298 0.979117i \(-0.565166\pi\)
−0.203298 + 0.979117i \(0.565166\pi\)
\(420\) 0 0
\(421\) 10.3228 0.503103 0.251551 0.967844i \(-0.419059\pi\)
0.251551 + 0.967844i \(0.419059\pi\)
\(422\) 0 0
\(423\) −9.00662 −0.437917
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 23.2136 1.12338
\(428\) 0 0
\(429\) 3.27316 0.158030
\(430\) 0 0
\(431\) 17.5662 0.846133 0.423066 0.906099i \(-0.360954\pi\)
0.423066 + 0.906099i \(0.360954\pi\)
\(432\) 0 0
\(433\) −7.00662 −0.336716 −0.168358 0.985726i \(-0.553847\pi\)
−0.168358 + 0.985726i \(0.553847\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 8.08604 0.386808
\(438\) 0 0
\(439\) −1.25993 −0.0601334 −0.0300667 0.999548i \(-0.509572\pi\)
−0.0300667 + 0.999548i \(0.509572\pi\)
\(440\) 0 0
\(441\) −0.733451 −0.0349262
\(442\) 0 0
\(443\) −26.8261 −1.27455 −0.637273 0.770638i \(-0.719938\pi\)
−0.637273 + 0.770638i \(0.719938\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −2.27978 −0.107830
\(448\) 0 0
\(449\) −32.7401 −1.54510 −0.772550 0.634954i \(-0.781019\pi\)
−0.772550 + 0.634954i \(0.781019\pi\)
\(450\) 0 0
\(451\) 25.3129 1.19194
\(452\) 0 0
\(453\) 6.46029 0.303531
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −33.5960 −1.57155 −0.785776 0.618511i \(-0.787736\pi\)
−0.785776 + 0.618511i \(0.787736\pi\)
\(458\) 0 0
\(459\) 4.50331 0.210196
\(460\) 0 0
\(461\) −41.8327 −1.94834 −0.974172 0.225807i \(-0.927498\pi\)
−0.974172 + 0.225807i \(0.927498\pi\)
\(462\) 0 0
\(463\) 6.05625 0.281458 0.140729 0.990048i \(-0.455055\pi\)
0.140729 + 0.990048i \(0.455055\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −27.2996 −1.26328 −0.631638 0.775263i \(-0.717617\pi\)
−0.631638 + 0.775263i \(0.717617\pi\)
\(468\) 0 0
\(469\) −38.9338 −1.79780
\(470\) 0 0
\(471\) −3.53971 −0.163101
\(472\) 0 0
\(473\) −13.0927 −0.602001
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −6.27978 −0.287531
\(478\) 0 0
\(479\) 28.2798 1.29214 0.646068 0.763280i \(-0.276412\pi\)
0.646068 + 0.763280i \(0.276412\pi\)
\(480\) 0 0
\(481\) 7.73345 0.352615
\(482\) 0 0
\(483\) −6.26655 −0.285138
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −24.9636 −1.13121 −0.565604 0.824677i \(-0.691357\pi\)
−0.565604 + 0.824677i \(0.691357\pi\)
\(488\) 0 0
\(489\) −18.7401 −0.847455
\(490\) 0 0
\(491\) 6.22353 0.280864 0.140432 0.990090i \(-0.455151\pi\)
0.140432 + 0.990090i \(0.455151\pi\)
\(492\) 0 0
\(493\) 35.0198 1.57721
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 14.3526 0.643802
\(498\) 0 0
\(499\) −19.2301 −0.860859 −0.430430 0.902624i \(-0.641638\pi\)
−0.430430 + 0.902624i \(0.641638\pi\)
\(500\) 0 0
\(501\) −4.00000 −0.178707
\(502\) 0 0
\(503\) −10.6970 −0.476958 −0.238479 0.971148i \(-0.576649\pi\)
−0.238479 + 0.971148i \(0.576649\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 0 0
\(509\) 0.740066 0.0328029 0.0164014 0.999865i \(-0.494779\pi\)
0.0164014 + 0.999865i \(0.494779\pi\)
\(510\) 0 0
\(511\) −39.4934 −1.74708
\(512\) 0 0
\(513\) −3.23014 −0.142614
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −29.4801 −1.29653
\(518\) 0 0
\(519\) −8.46029 −0.371365
\(520\) 0 0
\(521\) −35.5662 −1.55818 −0.779091 0.626911i \(-0.784319\pi\)
−0.779091 + 0.626911i \(0.784319\pi\)
\(522\) 0 0
\(523\) −14.0132 −0.612756 −0.306378 0.951910i \(-0.599117\pi\)
−0.306378 + 0.951910i \(0.599117\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 22.5463 0.982134
\(528\) 0 0
\(529\) −16.7335 −0.727541
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −7.73345 −0.334973
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 4.76986 0.205835
\(538\) 0 0
\(539\) −2.40071 −0.103406
\(540\) 0 0
\(541\) 11.7765 0.506310 0.253155 0.967426i \(-0.418532\pi\)
0.253155 + 0.967426i \(0.418532\pi\)
\(542\) 0 0
\(543\) −15.7335 −0.675187
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 4.00000 0.171028 0.0855138 0.996337i \(-0.472747\pi\)
0.0855138 + 0.996337i \(0.472747\pi\)
\(548\) 0 0
\(549\) 9.27316 0.395769
\(550\) 0 0
\(551\) −25.1191 −1.07011
\(552\) 0 0
\(553\) −11.8327 −0.503179
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 42.0265 1.78072 0.890359 0.455259i \(-0.150453\pi\)
0.890359 + 0.455259i \(0.150453\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) 14.7401 0.622326
\(562\) 0 0
\(563\) 25.3724 1.06932 0.534660 0.845067i \(-0.320440\pi\)
0.534660 + 0.845067i \(0.320440\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 2.50331 0.105129
\(568\) 0 0
\(569\) 36.0993 1.51336 0.756680 0.653785i \(-0.226820\pi\)
0.756680 + 0.653785i \(0.226820\pi\)
\(570\) 0 0
\(571\) −23.1871 −0.970351 −0.485175 0.874417i \(-0.661244\pi\)
−0.485175 + 0.874417i \(0.661244\pi\)
\(572\) 0 0
\(573\) 26.0132 1.08672
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −34.9636 −1.45555 −0.727777 0.685814i \(-0.759446\pi\)
−0.727777 + 0.685814i \(0.759446\pi\)
\(578\) 0 0
\(579\) −0.503308 −0.0209168
\(580\) 0 0
\(581\) 28.9206 1.19983
\(582\) 0 0
\(583\) −20.5548 −0.851291
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −18.9934 −0.783941 −0.391970 0.919978i \(-0.628206\pi\)
−0.391970 + 0.919978i \(0.628206\pi\)
\(588\) 0 0
\(589\) −16.1721 −0.666359
\(590\) 0 0
\(591\) −11.0066 −0.452752
\(592\) 0 0
\(593\) −10.5331 −0.432543 −0.216271 0.976333i \(-0.569390\pi\)
−0.216271 + 0.976333i \(0.569390\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −2.01323 −0.0823960
\(598\) 0 0
\(599\) −23.1059 −0.944081 −0.472040 0.881577i \(-0.656482\pi\)
−0.472040 + 0.881577i \(0.656482\pi\)
\(600\) 0 0
\(601\) −18.8129 −0.767393 −0.383697 0.923459i \(-0.625349\pi\)
−0.383697 + 0.923459i \(0.625349\pi\)
\(602\) 0 0
\(603\) −15.5529 −0.633365
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 5.09266 0.206705 0.103352 0.994645i \(-0.467043\pi\)
0.103352 + 0.994645i \(0.467043\pi\)
\(608\) 0 0
\(609\) 19.4669 0.788839
\(610\) 0 0
\(611\) 9.00662 0.364369
\(612\) 0 0
\(613\) −27.2004 −1.09861 −0.549306 0.835621i \(-0.685108\pi\)
−0.549306 + 0.835621i \(0.685108\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0.460287 0.0185304 0.00926522 0.999957i \(-0.497051\pi\)
0.00926522 + 0.999957i \(0.497051\pi\)
\(618\) 0 0
\(619\) −1.21691 −0.0489119 −0.0244559 0.999701i \(-0.507785\pi\)
−0.0244559 + 0.999701i \(0.507785\pi\)
\(620\) 0 0
\(621\) −2.50331 −0.100454
\(622\) 0 0
\(623\) −38.2665 −1.53312
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −10.5728 −0.422237
\(628\) 0 0
\(629\) 34.8261 1.38861
\(630\) 0 0
\(631\) 2.01323 0.0801454 0.0400727 0.999197i \(-0.487241\pi\)
0.0400727 + 0.999197i \(0.487241\pi\)
\(632\) 0 0
\(633\) 12.0000 0.476957
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0.733451 0.0290604
\(638\) 0 0
\(639\) 5.73345 0.226812
\(640\) 0 0
\(641\) −4.37424 −0.172772 −0.0863861 0.996262i \(-0.527532\pi\)
−0.0863861 + 0.996262i \(0.527532\pi\)
\(642\) 0 0
\(643\) −3.72022 −0.146711 −0.0733556 0.997306i \(-0.523371\pi\)
−0.0733556 + 0.997306i \(0.523371\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 12.4305 0.488693 0.244347 0.969688i \(-0.421427\pi\)
0.244347 + 0.969688i \(0.421427\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 12.5331 0.491211
\(652\) 0 0
\(653\) −22.0000 −0.860927 −0.430463 0.902608i \(-0.641650\pi\)
−0.430463 + 0.902608i \(0.641650\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −15.7765 −0.615499
\(658\) 0 0
\(659\) 29.3294 1.14251 0.571256 0.820772i \(-0.306456\pi\)
0.571256 + 0.820772i \(0.306456\pi\)
\(660\) 0 0
\(661\) 2.32280 0.0903465 0.0451732 0.998979i \(-0.485616\pi\)
0.0451732 + 0.998979i \(0.485616\pi\)
\(662\) 0 0
\(663\) −4.50331 −0.174894
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −19.4669 −0.753761
\(668\) 0 0
\(669\) 23.7897 0.919763
\(670\) 0 0
\(671\) 30.3526 1.17175
\(672\) 0 0
\(673\) 21.1059 0.813572 0.406786 0.913523i \(-0.366649\pi\)
0.406786 + 0.913523i \(0.366649\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 34.1937 1.31417 0.657086 0.753816i \(-0.271789\pi\)
0.657086 + 0.753816i \(0.271789\pi\)
\(678\) 0 0
\(679\) −31.2996 −1.20117
\(680\) 0 0
\(681\) −23.0198 −0.882122
\(682\) 0 0
\(683\) 32.4735 1.24256 0.621282 0.783587i \(-0.286612\pi\)
0.621282 + 0.783587i \(0.286612\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 7.77647 0.296691
\(688\) 0 0
\(689\) 6.27978 0.239241
\(690\) 0 0
\(691\) −34.8095 −1.32422 −0.662109 0.749408i \(-0.730338\pi\)
−0.662109 + 0.749408i \(0.730338\pi\)
\(692\) 0 0
\(693\) 8.19374 0.311254
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −34.8261 −1.31913
\(698\) 0 0
\(699\) 23.9702 0.906637
\(700\) 0 0
\(701\) 18.1507 0.685543 0.342772 0.939419i \(-0.388634\pi\)
0.342772 + 0.939419i \(0.388634\pi\)
\(702\) 0 0
\(703\) −24.9802 −0.942144
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 13.0927 0.492400
\(708\) 0 0
\(709\) −11.2434 −0.422254 −0.211127 0.977459i \(-0.567713\pi\)
−0.211127 + 0.977459i \(0.567713\pi\)
\(710\) 0 0
\(711\) −4.72684 −0.177270
\(712\) 0 0
\(713\) −12.5331 −0.469368
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −0.812878 −0.0303575
\(718\) 0 0
\(719\) −9.53971 −0.355771 −0.177886 0.984051i \(-0.556926\pi\)
−0.177886 + 0.984051i \(0.556926\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −20.0132 −0.744300
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −36.1125 −1.33934 −0.669669 0.742659i \(-0.733564\pi\)
−0.669669 + 0.742659i \(0.733564\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 18.0132 0.666243
\(732\) 0 0
\(733\) 34.2798 1.26615 0.633076 0.774089i \(-0.281792\pi\)
0.633076 + 0.774089i \(0.281792\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −50.9073 −1.87520
\(738\) 0 0
\(739\) 25.8625 0.951368 0.475684 0.879616i \(-0.342201\pi\)
0.475684 + 0.879616i \(0.342201\pi\)
\(740\) 0 0
\(741\) 3.23014 0.118662
\(742\) 0 0
\(743\) 4.55956 0.167274 0.0836370 0.996496i \(-0.473346\pi\)
0.0836370 + 0.996496i \(0.473346\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 11.5529 0.422700
\(748\) 0 0
\(749\) 1.81949 0.0664828
\(750\) 0 0
\(751\) 35.8327 1.30755 0.653777 0.756687i \(-0.273183\pi\)
0.653777 + 0.756687i \(0.273183\pi\)
\(752\) 0 0
\(753\) 11.3162 0.412385
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 29.4669 1.07099 0.535496 0.844538i \(-0.320125\pi\)
0.535496 + 0.844538i \(0.320125\pi\)
\(758\) 0 0
\(759\) −8.19374 −0.297414
\(760\) 0 0
\(761\) 22.4735 0.814664 0.407332 0.913280i \(-0.366459\pi\)
0.407332 + 0.913280i \(0.366459\pi\)
\(762\) 0 0
\(763\) 42.0132 1.52098
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −33.4669 −1.20685 −0.603424 0.797421i \(-0.706197\pi\)
−0.603424 + 0.797421i \(0.706197\pi\)
\(770\) 0 0
\(771\) 19.3294 0.696132
\(772\) 0 0
\(773\) 22.0860 0.794380 0.397190 0.917736i \(-0.369985\pi\)
0.397190 + 0.917736i \(0.369985\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 19.3592 0.694508
\(778\) 0 0
\(779\) 24.9802 0.895007
\(780\) 0 0
\(781\) 18.7665 0.671519
\(782\) 0 0
\(783\) 7.77647 0.277908
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 28.0000 0.998092 0.499046 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(788\) 0 0
\(789\) −12.2368 −0.435640
\(790\) 0 0
\(791\) 9.45367 0.336134
\(792\) 0 0
\(793\) −9.27316 −0.329300
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 11.1739 0.395800 0.197900 0.980222i \(-0.436588\pi\)
0.197900 + 0.980222i \(0.436588\pi\)
\(798\) 0 0
\(799\) 40.5596 1.43489
\(800\) 0 0
\(801\) −15.2864 −0.540118
\(802\) 0 0
\(803\) −51.6390 −1.82230
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 2.76986 0.0975036
\(808\) 0 0
\(809\) 18.0000 0.632846 0.316423 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) 0 0
\(811\) 4.68381 0.164471 0.0822355 0.996613i \(-0.473794\pi\)
0.0822355 + 0.996613i \(0.473794\pi\)
\(812\) 0 0
\(813\) 0.0860423 0.00301763
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −12.9206 −0.452034
\(818\) 0 0
\(819\) −2.50331 −0.0874726
\(820\) 0 0
\(821\) 14.8129 0.516973 0.258487 0.966015i \(-0.416776\pi\)
0.258487 + 0.966015i \(0.416776\pi\)
\(822\) 0 0
\(823\) −20.0265 −0.698079 −0.349039 0.937108i \(-0.613492\pi\)
−0.349039 + 0.937108i \(0.613492\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −41.9272 −1.45795 −0.728976 0.684540i \(-0.760003\pi\)
−0.728976 + 0.684540i \(0.760003\pi\)
\(828\) 0 0
\(829\) 30.0000 1.04194 0.520972 0.853574i \(-0.325570\pi\)
0.520972 + 0.853574i \(0.325570\pi\)
\(830\) 0 0
\(831\) 17.5529 0.608905
\(832\) 0 0
\(833\) 3.30295 0.114441
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 5.00662 0.173054
\(838\) 0 0
\(839\) 4.83934 0.167073 0.0835363 0.996505i \(-0.473379\pi\)
0.0835363 + 0.996505i \(0.473379\pi\)
\(840\) 0 0
\(841\) 31.4735 1.08529
\(842\) 0 0
\(843\) −24.0132 −0.827060
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −0.716935 −0.0246342
\(848\) 0 0
\(849\) 22.0132 0.755492
\(850\) 0 0
\(851\) −19.3592 −0.663625
\(852\) 0 0
\(853\) 34.6673 1.18698 0.593492 0.804840i \(-0.297749\pi\)
0.593492 + 0.804840i \(0.297749\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −22.6026 −0.772089 −0.386045 0.922480i \(-0.626159\pi\)
−0.386045 + 0.922480i \(0.626159\pi\)
\(858\) 0 0
\(859\) −8.16728 −0.278664 −0.139332 0.990246i \(-0.544496\pi\)
−0.139332 + 0.990246i \(0.544496\pi\)
\(860\) 0 0
\(861\) −19.3592 −0.659760
\(862\) 0 0
\(863\) −51.4934 −1.75285 −0.876427 0.481534i \(-0.840080\pi\)
−0.876427 + 0.481534i \(0.840080\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −3.27978 −0.111387
\(868\) 0 0
\(869\) −15.4717 −0.524842
\(870\) 0 0
\(871\) 15.5529 0.526991
\(872\) 0 0
\(873\) −12.5033 −0.423173
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 53.1323 1.79415 0.897076 0.441876i \(-0.145687\pi\)
0.897076 + 0.441876i \(0.145687\pi\)
\(878\) 0 0
\(879\) −8.99338 −0.303339
\(880\) 0 0
\(881\) 14.4471 0.486734 0.243367 0.969934i \(-0.421748\pi\)
0.243367 + 0.969934i \(0.421748\pi\)
\(882\) 0 0
\(883\) 1.98677 0.0668601 0.0334301 0.999441i \(-0.489357\pi\)
0.0334301 + 0.999441i \(0.489357\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 53.0761 1.78212 0.891060 0.453885i \(-0.149962\pi\)
0.891060 + 0.453885i \(0.149962\pi\)
\(888\) 0 0
\(889\) 16.3875 0.549618
\(890\) 0 0
\(891\) 3.27316 0.109655
\(892\) 0 0
\(893\) −29.0927 −0.973549
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 2.50331 0.0835830
\(898\) 0 0
\(899\) 38.9338 1.29852
\(900\) 0 0
\(901\) 28.2798 0.942136
\(902\) 0 0
\(903\) 10.0132 0.333219
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 7.46690 0.247934 0.123967 0.992286i \(-0.460438\pi\)
0.123967 + 0.992286i \(0.460438\pi\)
\(908\) 0 0
\(909\) 5.23014 0.173473
\(910\) 0 0
\(911\) 56.6456 1.87675 0.938376 0.345615i \(-0.112330\pi\)
0.938376 + 0.345615i \(0.112330\pi\)
\(912\) 0 0
\(913\) 37.8147 1.25148
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −28.1125 −0.928357
\(918\) 0 0
\(919\) −48.7533 −1.60822 −0.804111 0.594479i \(-0.797358\pi\)
−0.804111 + 0.594479i \(0.797358\pi\)
\(920\) 0 0
\(921\) −22.2930 −0.734580
\(922\) 0 0
\(923\) −5.73345 −0.188719
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −54.3923 −1.78455 −0.892276 0.451489i \(-0.850893\pi\)
−0.892276 + 0.451489i \(0.850893\pi\)
\(930\) 0 0
\(931\) −2.36915 −0.0776458
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −26.0000 −0.849383 −0.424691 0.905338i \(-0.639617\pi\)
−0.424691 + 0.905338i \(0.639617\pi\)
\(938\) 0 0
\(939\) −17.5529 −0.572819
\(940\) 0 0
\(941\) −32.8526 −1.07096 −0.535482 0.844547i \(-0.679870\pi\)
−0.535482 + 0.844547i \(0.679870\pi\)
\(942\) 0 0
\(943\) 19.3592 0.630423
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 6.63237 0.215523 0.107762 0.994177i \(-0.465632\pi\)
0.107762 + 0.994177i \(0.465632\pi\)
\(948\) 0 0
\(949\) 15.7765 0.512126
\(950\) 0 0
\(951\) −4.54633 −0.147425
\(952\) 0 0
\(953\) −40.5298 −1.31289 −0.656444 0.754375i \(-0.727940\pi\)
−0.656444 + 0.754375i \(0.727940\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 25.4537 0.822800
\(958\) 0 0
\(959\) 33.9272 1.09557
\(960\) 0 0
\(961\) −5.93380 −0.191413
\(962\) 0 0
\(963\) 0.726836 0.0234220
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −43.3426 −1.39381 −0.696903 0.717166i \(-0.745439\pi\)
−0.696903 + 0.717166i \(0.745439\pi\)
\(968\) 0 0
\(969\) 14.5463 0.467295
\(970\) 0 0
\(971\) 3.78970 0.121617 0.0608087 0.998149i \(-0.480632\pi\)
0.0608087 + 0.998149i \(0.480632\pi\)
\(972\) 0 0
\(973\) −54.4055 −1.74416
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −17.4669 −0.558816 −0.279408 0.960173i \(-0.590138\pi\)
−0.279408 + 0.960173i \(0.590138\pi\)
\(978\) 0 0
\(979\) −50.0349 −1.59912
\(980\) 0 0
\(981\) 16.7831 0.535843
\(982\) 0 0
\(983\) 16.0265 0.511165 0.255582 0.966787i \(-0.417733\pi\)
0.255582 + 0.966787i \(0.417733\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 22.5463 0.717658
\(988\) 0 0
\(989\) −10.0132 −0.318402
\(990\) 0 0
\(991\) −36.3923 −1.15604 −0.578019 0.816023i \(-0.696174\pi\)
−0.578019 + 0.816023i \(0.696174\pi\)
\(992\) 0 0
\(993\) −19.7897 −0.628007
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 19.0331 0.602784 0.301392 0.953500i \(-0.402549\pi\)
0.301392 + 0.953500i \(0.402549\pi\)
\(998\) 0 0
\(999\) 7.73345 0.244676
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 7800.2.a.bf.1.3 3
5.4 even 2 1560.2.a.p.1.1 3
15.14 odd 2 4680.2.a.bl.1.1 3
20.19 odd 2 3120.2.a.bh.1.3 3
60.59 even 2 9360.2.a.db.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1560.2.a.p.1.1 3 5.4 even 2
3120.2.a.bh.1.3 3 20.19 odd 2
4680.2.a.bl.1.1 3 15.14 odd 2
7800.2.a.bf.1.3 3 1.1 even 1 trivial
9360.2.a.db.1.3 3 60.59 even 2