Properties

Label 4200.2.a.bp.1.2
Level $4200$
Weight $2$
Character 4200.1
Self dual yes
Analytic conductor $33.537$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4200 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4200.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(33.5371688489\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 840)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.17009\) of defining polynomial
Character \(\chi\) \(=\) 4200.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} -3.26180 q^{11} +0.340173 q^{13} +5.75872 q^{17} -6.49693 q^{19} +1.00000 q^{21} -8.49693 q^{23} +1.00000 q^{27} -2.00000 q^{29} -8.34017 q^{31} -3.26180 q^{33} -6.15676 q^{37} +0.340173 q^{39} +0.340173 q^{41} -8.68035 q^{43} +1.00000 q^{49} +5.75872 q^{51} -8.34017 q^{53} -6.49693 q^{57} +6.83710 q^{59} +15.3607 q^{61} +1.00000 q^{63} +14.8371 q^{67} -8.49693 q^{69} -15.9421 q^{71} -1.50307 q^{73} -3.26180 q^{77} +8.68035 q^{79} +1.00000 q^{81} +6.83710 q^{83} -2.00000 q^{87} -15.1773 q^{89} +0.340173 q^{91} -8.34017 q^{93} -6.49693 q^{97} -3.26180 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{3} + 3 q^{7} + 3 q^{9} - 2 q^{11} - 10 q^{13} - 8 q^{17} - 2 q^{19} + 3 q^{21} - 8 q^{23} + 3 q^{27} - 6 q^{29} - 14 q^{31} - 2 q^{33} - 12 q^{37} - 10 q^{39} - 10 q^{41} - 4 q^{43} + 3 q^{49} - 8 q^{51} - 14 q^{53} - 2 q^{57} - 8 q^{59} + 2 q^{61} + 3 q^{63} + 16 q^{67} - 8 q^{69} - 18 q^{71} - 22 q^{73} - 2 q^{77} + 4 q^{79} + 3 q^{81} - 8 q^{83} - 6 q^{87} - 6 q^{89} - 10 q^{91} - 14 q^{93} - 2 q^{97} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.26180 −0.983468 −0.491734 0.870745i \(-0.663637\pi\)
−0.491734 + 0.870745i \(0.663637\pi\)
\(12\) 0 0
\(13\) 0.340173 0.0943470 0.0471735 0.998887i \(-0.484979\pi\)
0.0471735 + 0.998887i \(0.484979\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.75872 1.39670 0.698348 0.715759i \(-0.253919\pi\)
0.698348 + 0.715759i \(0.253919\pi\)
\(18\) 0 0
\(19\) −6.49693 −1.49050 −0.745249 0.666786i \(-0.767669\pi\)
−0.745249 + 0.666786i \(0.767669\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −8.49693 −1.77173 −0.885866 0.463941i \(-0.846435\pi\)
−0.885866 + 0.463941i \(0.846435\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −8.34017 −1.49794 −0.748970 0.662604i \(-0.769451\pi\)
−0.748970 + 0.662604i \(0.769451\pi\)
\(32\) 0 0
\(33\) −3.26180 −0.567806
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6.15676 −1.01216 −0.506082 0.862485i \(-0.668907\pi\)
−0.506082 + 0.862485i \(0.668907\pi\)
\(38\) 0 0
\(39\) 0.340173 0.0544713
\(40\) 0 0
\(41\) 0.340173 0.0531261 0.0265630 0.999647i \(-0.491544\pi\)
0.0265630 + 0.999647i \(0.491544\pi\)
\(42\) 0 0
\(43\) −8.68035 −1.32374 −0.661870 0.749618i \(-0.730237\pi\)
−0.661870 + 0.749618i \(0.730237\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 5.75872 0.806383
\(52\) 0 0
\(53\) −8.34017 −1.14561 −0.572805 0.819691i \(-0.694145\pi\)
−0.572805 + 0.819691i \(0.694145\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −6.49693 −0.860539
\(58\) 0 0
\(59\) 6.83710 0.890115 0.445057 0.895502i \(-0.353183\pi\)
0.445057 + 0.895502i \(0.353183\pi\)
\(60\) 0 0
\(61\) 15.3607 1.96674 0.983368 0.181627i \(-0.0581363\pi\)
0.983368 + 0.181627i \(0.0581363\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 14.8371 1.81264 0.906320 0.422592i \(-0.138880\pi\)
0.906320 + 0.422592i \(0.138880\pi\)
\(68\) 0 0
\(69\) −8.49693 −1.02291
\(70\) 0 0
\(71\) −15.9421 −1.89198 −0.945992 0.324190i \(-0.894908\pi\)
−0.945992 + 0.324190i \(0.894908\pi\)
\(72\) 0 0
\(73\) −1.50307 −0.175921 −0.0879606 0.996124i \(-0.528035\pi\)
−0.0879606 + 0.996124i \(0.528035\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −3.26180 −0.371716
\(78\) 0 0
\(79\) 8.68035 0.976615 0.488308 0.872672i \(-0.337614\pi\)
0.488308 + 0.872672i \(0.337614\pi\)
\(80\) 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
\(87\) −2.00000 −0.214423
\(88\) 0 0
\(89\) −15.1773 −1.60879 −0.804394 0.594096i \(-0.797510\pi\)
−0.804394 + 0.594096i \(0.797510\pi\)
\(90\) 0 0
\(91\) 0.340173 0.0356598
\(92\) 0 0
\(93\) −8.34017 −0.864836
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −6.49693 −0.659663 −0.329832 0.944040i \(-0.606992\pi\)
−0.329832 + 0.944040i \(0.606992\pi\)
\(98\) 0 0
\(99\) −3.26180 −0.327823
\(100\) 0 0
\(101\) −2.18342 −0.217258 −0.108629 0.994082i \(-0.534646\pi\)
−0.108629 + 0.994082i \(0.534646\pi\)
\(102\) 0 0
\(103\) −5.84324 −0.575752 −0.287876 0.957668i \(-0.592949\pi\)
−0.287876 + 0.957668i \(0.592949\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −8.18342 −0.791121 −0.395560 0.918440i \(-0.629450\pi\)
−0.395560 + 0.918440i \(0.629450\pi\)
\(108\) 0 0
\(109\) 16.8371 1.61270 0.806351 0.591437i \(-0.201439\pi\)
0.806351 + 0.591437i \(0.201439\pi\)
\(110\) 0 0
\(111\) −6.15676 −0.584373
\(112\) 0 0
\(113\) −13.0205 −1.22487 −0.612434 0.790522i \(-0.709809\pi\)
−0.612434 + 0.790522i \(0.709809\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0.340173 0.0314490
\(118\) 0 0
\(119\) 5.75872 0.527901
\(120\) 0 0
\(121\) −0.360692 −0.0327902
\(122\) 0 0
\(123\) 0.340173 0.0306724
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −1.84324 −0.163562 −0.0817808 0.996650i \(-0.526061\pi\)
−0.0817808 + 0.996650i \(0.526061\pi\)
\(128\) 0 0
\(129\) −8.68035 −0.764262
\(130\) 0 0
\(131\) −0.313511 −0.0273916 −0.0136958 0.999906i \(-0.504360\pi\)
−0.0136958 + 0.999906i \(0.504360\pi\)
\(132\) 0 0
\(133\) −6.49693 −0.563355
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2.18342 −0.186542 −0.0932710 0.995641i \(-0.529732\pi\)
−0.0932710 + 0.995641i \(0.529732\pi\)
\(138\) 0 0
\(139\) −1.02052 −0.0865593 −0.0432796 0.999063i \(-0.513781\pi\)
−0.0432796 + 0.999063i \(0.513781\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −1.10957 −0.0927873
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) 14.6803 1.20266 0.601330 0.799000i \(-0.294638\pi\)
0.601330 + 0.799000i \(0.294638\pi\)
\(150\) 0 0
\(151\) 2.15676 0.175514 0.0877571 0.996142i \(-0.472030\pi\)
0.0877571 + 0.996142i \(0.472030\pi\)
\(152\) 0 0
\(153\) 5.75872 0.465565
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −6.18342 −0.493490 −0.246745 0.969080i \(-0.579361\pi\)
−0.246745 + 0.969080i \(0.579361\pi\)
\(158\) 0 0
\(159\) −8.34017 −0.661419
\(160\) 0 0
\(161\) −8.49693 −0.669652
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.47641 0.114248 0.0571240 0.998367i \(-0.481807\pi\)
0.0571240 + 0.998367i \(0.481807\pi\)
\(168\) 0 0
\(169\) −12.8843 −0.991099
\(170\) 0 0
\(171\) −6.49693 −0.496833
\(172\) 0 0
\(173\) 1.75872 0.133713 0.0668566 0.997763i \(-0.478703\pi\)
0.0668566 + 0.997763i \(0.478703\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 6.83710 0.513908
\(178\) 0 0
\(179\) 0.424694 0.0317431 0.0158716 0.999874i \(-0.494948\pi\)
0.0158716 + 0.999874i \(0.494948\pi\)
\(180\) 0 0
\(181\) 10.3668 0.770561 0.385280 0.922800i \(-0.374105\pi\)
0.385280 + 0.922800i \(0.374105\pi\)
\(182\) 0 0
\(183\) 15.3607 1.13550
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −18.7838 −1.37361
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −21.7321 −1.57248 −0.786238 0.617923i \(-0.787974\pi\)
−0.786238 + 0.617923i \(0.787974\pi\)
\(192\) 0 0
\(193\) 8.36683 0.602258 0.301129 0.953583i \(-0.402637\pi\)
0.301129 + 0.953583i \(0.402637\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −13.3340 −0.950010 −0.475005 0.879983i \(-0.657554\pi\)
−0.475005 + 0.879983i \(0.657554\pi\)
\(198\) 0 0
\(199\) 6.49693 0.460555 0.230278 0.973125i \(-0.426037\pi\)
0.230278 + 0.973125i \(0.426037\pi\)
\(200\) 0 0
\(201\) 14.8371 1.04653
\(202\) 0 0
\(203\) −2.00000 −0.140372
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −8.49693 −0.590577
\(208\) 0 0
\(209\) 21.1917 1.46586
\(210\) 0 0
\(211\) 6.83710 0.470685 0.235343 0.971912i \(-0.424379\pi\)
0.235343 + 0.971912i \(0.424379\pi\)
\(212\) 0 0
\(213\) −15.9421 −1.09234
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −8.34017 −0.566168
\(218\) 0 0
\(219\) −1.50307 −0.101568
\(220\) 0 0
\(221\) 1.95896 0.131774
\(222\) 0 0
\(223\) −17.3607 −1.16256 −0.581279 0.813704i \(-0.697447\pi\)
−0.581279 + 0.813704i \(0.697447\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −10.1568 −0.674128 −0.337064 0.941482i \(-0.609434\pi\)
−0.337064 + 0.941482i \(0.609434\pi\)
\(228\) 0 0
\(229\) 14.9939 0.990822 0.495411 0.868659i \(-0.335017\pi\)
0.495411 + 0.868659i \(0.335017\pi\)
\(230\) 0 0
\(231\) −3.26180 −0.214610
\(232\) 0 0
\(233\) 11.5441 0.756280 0.378140 0.925748i \(-0.376564\pi\)
0.378140 + 0.925748i \(0.376564\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 8.68035 0.563849
\(238\) 0 0
\(239\) −10.8950 −0.704736 −0.352368 0.935861i \(-0.614624\pi\)
−0.352368 + 0.935861i \(0.614624\pi\)
\(240\) 0 0
\(241\) −6.68035 −0.430319 −0.215159 0.976579i \(-0.569027\pi\)
−0.215159 + 0.976579i \(0.569027\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −2.21008 −0.140624
\(248\) 0 0
\(249\) 6.83710 0.433284
\(250\) 0 0
\(251\) 24.1978 1.52735 0.763676 0.645600i \(-0.223393\pi\)
0.763676 + 0.645600i \(0.223393\pi\)
\(252\) 0 0
\(253\) 27.7152 1.74244
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −15.2351 −0.950342 −0.475171 0.879894i \(-0.657614\pi\)
−0.475171 + 0.879894i \(0.657614\pi\)
\(258\) 0 0
\(259\) −6.15676 −0.382562
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) 0 0
\(263\) −29.1773 −1.79915 −0.899574 0.436769i \(-0.856123\pi\)
−0.899574 + 0.436769i \(0.856123\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −15.1773 −0.928834
\(268\) 0 0
\(269\) −6.13009 −0.373758 −0.186879 0.982383i \(-0.559837\pi\)
−0.186879 + 0.982383i \(0.559837\pi\)
\(270\) 0 0
\(271\) 25.7009 1.56122 0.780608 0.625021i \(-0.214909\pi\)
0.780608 + 0.625021i \(0.214909\pi\)
\(272\) 0 0
\(273\) 0.340173 0.0205882
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −11.2039 −0.673179 −0.336590 0.941651i \(-0.609274\pi\)
−0.336590 + 0.941651i \(0.609274\pi\)
\(278\) 0 0
\(279\) −8.34017 −0.499313
\(280\) 0 0
\(281\) 24.3545 1.45287 0.726435 0.687235i \(-0.241176\pi\)
0.726435 + 0.687235i \(0.241176\pi\)
\(282\) 0 0
\(283\) 6.15676 0.365981 0.182991 0.983115i \(-0.441422\pi\)
0.182991 + 0.983115i \(0.441422\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0.340173 0.0200798
\(288\) 0 0
\(289\) 16.1629 0.950759
\(290\) 0 0
\(291\) −6.49693 −0.380857
\(292\) 0 0
\(293\) −1.75872 −0.102746 −0.0513729 0.998680i \(-0.516360\pi\)
−0.0513729 + 0.998680i \(0.516360\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −3.26180 −0.189269
\(298\) 0 0
\(299\) −2.89043 −0.167158
\(300\) 0 0
\(301\) −8.68035 −0.500327
\(302\) 0 0
\(303\) −2.18342 −0.125434
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 27.2039 1.55261 0.776305 0.630357i \(-0.217092\pi\)
0.776305 + 0.630357i \(0.217092\pi\)
\(308\) 0 0
\(309\) −5.84324 −0.332411
\(310\) 0 0
\(311\) −21.8432 −1.23862 −0.619308 0.785148i \(-0.712587\pi\)
−0.619308 + 0.785148i \(0.712587\pi\)
\(312\) 0 0
\(313\) −25.3874 −1.43498 −0.717489 0.696570i \(-0.754709\pi\)
−0.717489 + 0.696570i \(0.754709\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 22.6947 1.27466 0.637331 0.770590i \(-0.280038\pi\)
0.637331 + 0.770590i \(0.280038\pi\)
\(318\) 0 0
\(319\) 6.52359 0.365251
\(320\) 0 0
\(321\) −8.18342 −0.456754
\(322\) 0 0
\(323\) −37.4140 −2.08177
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 16.8371 0.931094
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 5.47641 0.301011 0.150505 0.988609i \(-0.451910\pi\)
0.150505 + 0.988609i \(0.451910\pi\)
\(332\) 0 0
\(333\) −6.15676 −0.337388
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 13.6742 0.744881 0.372441 0.928056i \(-0.378521\pi\)
0.372441 + 0.928056i \(0.378521\pi\)
\(338\) 0 0
\(339\) −13.0205 −0.707178
\(340\) 0 0
\(341\) 27.2039 1.47318
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −0.183417 −0.00984637 −0.00492318 0.999988i \(-0.501567\pi\)
−0.00492318 + 0.999988i \(0.501567\pi\)
\(348\) 0 0
\(349\) −7.67420 −0.410791 −0.205395 0.978679i \(-0.565848\pi\)
−0.205395 + 0.978679i \(0.565848\pi\)
\(350\) 0 0
\(351\) 0.340173 0.0181571
\(352\) 0 0
\(353\) −19.4329 −1.03431 −0.517155 0.855892i \(-0.673009\pi\)
−0.517155 + 0.855892i \(0.673009\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 5.75872 0.304784
\(358\) 0 0
\(359\) −25.4186 −1.34154 −0.670770 0.741666i \(-0.734036\pi\)
−0.670770 + 0.741666i \(0.734036\pi\)
\(360\) 0 0
\(361\) 23.2101 1.22158
\(362\) 0 0
\(363\) −0.360692 −0.0189314
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −9.36069 −0.488624 −0.244312 0.969697i \(-0.578562\pi\)
−0.244312 + 0.969697i \(0.578562\pi\)
\(368\) 0 0
\(369\) 0.340173 0.0177087
\(370\) 0 0
\(371\) −8.34017 −0.433000
\(372\) 0 0
\(373\) −27.5174 −1.42480 −0.712400 0.701774i \(-0.752392\pi\)
−0.712400 + 0.701774i \(0.752392\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −0.680346 −0.0350396
\(378\) 0 0
\(379\) 9.84324 0.505614 0.252807 0.967517i \(-0.418646\pi\)
0.252807 + 0.967517i \(0.418646\pi\)
\(380\) 0 0
\(381\) −1.84324 −0.0944323
\(382\) 0 0
\(383\) 27.8310 1.42210 0.711048 0.703144i \(-0.248221\pi\)
0.711048 + 0.703144i \(0.248221\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −8.68035 −0.441247
\(388\) 0 0
\(389\) −3.67420 −0.186289 −0.0931447 0.995653i \(-0.529692\pi\)
−0.0931447 + 0.995653i \(0.529692\pi\)
\(390\) 0 0
\(391\) −48.9315 −2.47457
\(392\) 0 0
\(393\) −0.313511 −0.0158145
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −22.8638 −1.14750 −0.573750 0.819031i \(-0.694512\pi\)
−0.573750 + 0.819031i \(0.694512\pi\)
\(398\) 0 0
\(399\) −6.49693 −0.325253
\(400\) 0 0
\(401\) 5.31965 0.265651 0.132825 0.991139i \(-0.457595\pi\)
0.132825 + 0.991139i \(0.457595\pi\)
\(402\) 0 0
\(403\) −2.83710 −0.141326
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 20.0821 0.995432
\(408\) 0 0
\(409\) 27.7275 1.37104 0.685519 0.728055i \(-0.259575\pi\)
0.685519 + 0.728055i \(0.259575\pi\)
\(410\) 0 0
\(411\) −2.18342 −0.107700
\(412\) 0 0
\(413\) 6.83710 0.336432
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −1.02052 −0.0499750
\(418\) 0 0
\(419\) −23.5174 −1.14890 −0.574451 0.818539i \(-0.694785\pi\)
−0.574451 + 0.818539i \(0.694785\pi\)
\(420\) 0 0
\(421\) −1.15061 −0.0560774 −0.0280387 0.999607i \(-0.508926\pi\)
−0.0280387 + 0.999607i \(0.508926\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 15.3607 0.743356
\(428\) 0 0
\(429\) −1.10957 −0.0535708
\(430\) 0 0
\(431\) 17.4186 0.839022 0.419511 0.907750i \(-0.362202\pi\)
0.419511 + 0.907750i \(0.362202\pi\)
\(432\) 0 0
\(433\) −26.0144 −1.25017 −0.625086 0.780556i \(-0.714936\pi\)
−0.625086 + 0.780556i \(0.714936\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 55.2039 2.64076
\(438\) 0 0
\(439\) 5.13624 0.245139 0.122570 0.992460i \(-0.460887\pi\)
0.122570 + 0.992460i \(0.460887\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 18.5380 0.880765 0.440383 0.897810i \(-0.354843\pi\)
0.440383 + 0.897810i \(0.354843\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 14.6803 0.694357
\(448\) 0 0
\(449\) −10.3135 −0.486725 −0.243362 0.969935i \(-0.578250\pi\)
−0.243362 + 0.969935i \(0.578250\pi\)
\(450\) 0 0
\(451\) −1.10957 −0.0522478
\(452\) 0 0
\(453\) 2.15676 0.101333
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 4.36683 0.204272 0.102136 0.994770i \(-0.467432\pi\)
0.102136 + 0.994770i \(0.467432\pi\)
\(458\) 0 0
\(459\) 5.75872 0.268794
\(460\) 0 0
\(461\) −33.7009 −1.56961 −0.784803 0.619745i \(-0.787236\pi\)
−0.784803 + 0.619745i \(0.787236\pi\)
\(462\) 0 0
\(463\) −10.4703 −0.486595 −0.243297 0.969952i \(-0.578229\pi\)
−0.243297 + 0.969952i \(0.578229\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −3.51745 −0.162768 −0.0813840 0.996683i \(-0.525934\pi\)
−0.0813840 + 0.996683i \(0.525934\pi\)
\(468\) 0 0
\(469\) 14.8371 0.685114
\(470\) 0 0
\(471\) −6.18342 −0.284917
\(472\) 0 0
\(473\) 28.3135 1.30186
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −8.34017 −0.381870
\(478\) 0 0
\(479\) −21.8432 −0.998043 −0.499022 0.866590i \(-0.666307\pi\)
−0.499022 + 0.866590i \(0.666307\pi\)
\(480\) 0 0
\(481\) −2.09436 −0.0954947
\(482\) 0 0
\(483\) −8.49693 −0.386624
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 7.88428 0.357271 0.178635 0.983915i \(-0.442832\pi\)
0.178635 + 0.983915i \(0.442832\pi\)
\(488\) 0 0
\(489\) −4.00000 −0.180886
\(490\) 0 0
\(491\) 19.7731 0.892347 0.446174 0.894946i \(-0.352786\pi\)
0.446174 + 0.894946i \(0.352786\pi\)
\(492\) 0 0
\(493\) −11.5174 −0.518720
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −15.9421 −0.715103
\(498\) 0 0
\(499\) 39.5174 1.76904 0.884522 0.466499i \(-0.154485\pi\)
0.884522 + 0.466499i \(0.154485\pi\)
\(500\) 0 0
\(501\) 1.47641 0.0659611
\(502\) 0 0
\(503\) −11.2039 −0.499559 −0.249779 0.968303i \(-0.580358\pi\)
−0.249779 + 0.968303i \(0.580358\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −12.8843 −0.572211
\(508\) 0 0
\(509\) −23.5441 −1.04357 −0.521787 0.853076i \(-0.674734\pi\)
−0.521787 + 0.853076i \(0.674734\pi\)
\(510\) 0 0
\(511\) −1.50307 −0.0664920
\(512\) 0 0
\(513\) −6.49693 −0.286846
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 1.75872 0.0771994
\(520\) 0 0
\(521\) 24.6537 1.08010 0.540049 0.841634i \(-0.318406\pi\)
0.540049 + 0.841634i \(0.318406\pi\)
\(522\) 0 0
\(523\) −2.63931 −0.115409 −0.0577044 0.998334i \(-0.518378\pi\)
−0.0577044 + 0.998334i \(0.518378\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −48.0288 −2.09217
\(528\) 0 0
\(529\) 49.1978 2.13903
\(530\) 0 0
\(531\) 6.83710 0.296705
\(532\) 0 0
\(533\) 0.115718 0.00501229
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0.424694 0.0183269
\(538\) 0 0
\(539\) −3.26180 −0.140495
\(540\) 0 0
\(541\) 25.1506 1.08131 0.540655 0.841245i \(-0.318177\pi\)
0.540655 + 0.841245i \(0.318177\pi\)
\(542\) 0 0
\(543\) 10.3668 0.444883
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −35.1506 −1.50293 −0.751466 0.659772i \(-0.770653\pi\)
−0.751466 + 0.659772i \(0.770653\pi\)
\(548\) 0 0
\(549\) 15.3607 0.655578
\(550\) 0 0
\(551\) 12.9939 0.553557
\(552\) 0 0
\(553\) 8.68035 0.369126
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 26.3812 1.11781 0.558904 0.829232i \(-0.311222\pi\)
0.558904 + 0.829232i \(0.311222\pi\)
\(558\) 0 0
\(559\) −2.95282 −0.124891
\(560\) 0 0
\(561\) −18.7838 −0.793052
\(562\) 0 0
\(563\) 12.9939 0.547626 0.273813 0.961783i \(-0.411715\pi\)
0.273813 + 0.961783i \(0.411715\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −6.36683 −0.266912 −0.133456 0.991055i \(-0.542607\pi\)
−0.133456 + 0.991055i \(0.542607\pi\)
\(570\) 0 0
\(571\) 2.63931 0.110452 0.0552258 0.998474i \(-0.482412\pi\)
0.0552258 + 0.998474i \(0.482412\pi\)
\(572\) 0 0
\(573\) −21.7321 −0.907870
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 33.3340 1.38771 0.693857 0.720113i \(-0.255910\pi\)
0.693857 + 0.720113i \(0.255910\pi\)
\(578\) 0 0
\(579\) 8.36683 0.347714
\(580\) 0 0
\(581\) 6.83710 0.283651
\(582\) 0 0
\(583\) 27.2039 1.12667
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −17.3074 −0.714352 −0.357176 0.934037i \(-0.616260\pi\)
−0.357176 + 0.934037i \(0.616260\pi\)
\(588\) 0 0
\(589\) 54.1855 2.23267
\(590\) 0 0
\(591\) −13.3340 −0.548489
\(592\) 0 0
\(593\) 47.1194 1.93496 0.967481 0.252943i \(-0.0813984\pi\)
0.967481 + 0.252943i \(0.0813984\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 6.49693 0.265902
\(598\) 0 0
\(599\) −13.1050 −0.535457 −0.267729 0.963494i \(-0.586273\pi\)
−0.267729 + 0.963494i \(0.586273\pi\)
\(600\) 0 0
\(601\) 29.0349 1.18436 0.592179 0.805806i \(-0.298268\pi\)
0.592179 + 0.805806i \(0.298268\pi\)
\(602\) 0 0
\(603\) 14.8371 0.604213
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 25.9877 1.05481 0.527404 0.849614i \(-0.323165\pi\)
0.527404 + 0.849614i \(0.323165\pi\)
\(608\) 0 0
\(609\) −2.00000 −0.0810441
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 20.1978 0.815781 0.407891 0.913031i \(-0.366264\pi\)
0.407891 + 0.913031i \(0.366264\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 47.8576 1.92668 0.963338 0.268292i \(-0.0864592\pi\)
0.963338 + 0.268292i \(0.0864592\pi\)
\(618\) 0 0
\(619\) −18.3812 −0.738803 −0.369402 0.929270i \(-0.620437\pi\)
−0.369402 + 0.929270i \(0.620437\pi\)
\(620\) 0 0
\(621\) −8.49693 −0.340970
\(622\) 0 0
\(623\) −15.1773 −0.608065
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 21.1917 0.846313
\(628\) 0 0
\(629\) −35.4551 −1.41369
\(630\) 0 0
\(631\) −7.94668 −0.316352 −0.158176 0.987411i \(-0.550561\pi\)
−0.158176 + 0.987411i \(0.550561\pi\)
\(632\) 0 0
\(633\) 6.83710 0.271750
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0.340173 0.0134781
\(638\) 0 0
\(639\) −15.9421 −0.630661
\(640\) 0 0
\(641\) 15.7275 0.621200 0.310600 0.950541i \(-0.399470\pi\)
0.310600 + 0.950541i \(0.399470\pi\)
\(642\) 0 0
\(643\) 44.5646 1.75746 0.878729 0.477322i \(-0.158392\pi\)
0.878729 + 0.477322i \(0.158392\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −5.16290 −0.202974 −0.101487 0.994837i \(-0.532360\pi\)
−0.101487 + 0.994837i \(0.532360\pi\)
\(648\) 0 0
\(649\) −22.3012 −0.875400
\(650\) 0 0
\(651\) −8.34017 −0.326877
\(652\) 0 0
\(653\) −7.49079 −0.293137 −0.146569 0.989201i \(-0.546823\pi\)
−0.146569 + 0.989201i \(0.546823\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −1.50307 −0.0586404
\(658\) 0 0
\(659\) −22.7259 −0.885276 −0.442638 0.896700i \(-0.645957\pi\)
−0.442638 + 0.896700i \(0.645957\pi\)
\(660\) 0 0
\(661\) −12.3258 −0.479418 −0.239709 0.970845i \(-0.577052\pi\)
−0.239709 + 0.970845i \(0.577052\pi\)
\(662\) 0 0
\(663\) 1.95896 0.0760798
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 16.9939 0.658005
\(668\) 0 0
\(669\) −17.3607 −0.671203
\(670\) 0 0
\(671\) −50.1034 −1.93422
\(672\) 0 0
\(673\) 25.6742 0.989668 0.494834 0.868988i \(-0.335229\pi\)
0.494834 + 0.868988i \(0.335229\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −29.2762 −1.12517 −0.562587 0.826738i \(-0.690194\pi\)
−0.562587 + 0.826738i \(0.690194\pi\)
\(678\) 0 0
\(679\) −6.49693 −0.249329
\(680\) 0 0
\(681\) −10.1568 −0.389208
\(682\) 0 0
\(683\) −25.5441 −0.977418 −0.488709 0.872447i \(-0.662532\pi\)
−0.488709 + 0.872447i \(0.662532\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 14.9939 0.572051
\(688\) 0 0
\(689\) −2.83710 −0.108085
\(690\) 0 0
\(691\) 26.3812 1.00359 0.501794 0.864987i \(-0.332673\pi\)
0.501794 + 0.864987i \(0.332673\pi\)
\(692\) 0 0
\(693\) −3.26180 −0.123905
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 1.95896 0.0742010
\(698\) 0 0
\(699\) 11.5441 0.436638
\(700\) 0 0
\(701\) −23.6742 −0.894162 −0.447081 0.894493i \(-0.647536\pi\)
−0.447081 + 0.894493i \(0.647536\pi\)
\(702\) 0 0
\(703\) 40.0000 1.50863
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −2.18342 −0.0821159
\(708\) 0 0
\(709\) −40.7214 −1.52932 −0.764662 0.644432i \(-0.777094\pi\)
−0.764662 + 0.644432i \(0.777094\pi\)
\(710\) 0 0
\(711\) 8.68035 0.325538
\(712\) 0 0
\(713\) 70.8659 2.65395
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −10.8950 −0.406880
\(718\) 0 0
\(719\) 22.3545 0.833684 0.416842 0.908979i \(-0.363137\pi\)
0.416842 + 0.908979i \(0.363137\pi\)
\(720\) 0 0
\(721\) −5.84324 −0.217614
\(722\) 0 0
\(723\) −6.68035 −0.248445
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 5.10957 0.189504 0.0947518 0.995501i \(-0.469794\pi\)
0.0947518 + 0.995501i \(0.469794\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −49.9877 −1.84886
\(732\) 0 0
\(733\) −41.6475 −1.53829 −0.769144 0.639076i \(-0.779317\pi\)
−0.769144 + 0.639076i \(0.779317\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −48.3956 −1.78267
\(738\) 0 0
\(739\) 32.3135 1.18867 0.594336 0.804217i \(-0.297415\pi\)
0.594336 + 0.804217i \(0.297415\pi\)
\(740\) 0 0
\(741\) −2.21008 −0.0811893
\(742\) 0 0
\(743\) −30.1711 −1.10687 −0.553436 0.832892i \(-0.686684\pi\)
−0.553436 + 0.832892i \(0.686684\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 6.83710 0.250156
\(748\) 0 0
\(749\) −8.18342 −0.299016
\(750\) 0 0
\(751\) −46.1855 −1.68533 −0.842667 0.538436i \(-0.819015\pi\)
−0.842667 + 0.538436i \(0.819015\pi\)
\(752\) 0 0
\(753\) 24.1978 0.881817
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −50.5523 −1.83736 −0.918678 0.395007i \(-0.870742\pi\)
−0.918678 + 0.395007i \(0.870742\pi\)
\(758\) 0 0
\(759\) 27.7152 1.00600
\(760\) 0 0
\(761\) −12.1711 −0.441203 −0.220602 0.975364i \(-0.570802\pi\)
−0.220602 + 0.975364i \(0.570802\pi\)
\(762\) 0 0
\(763\) 16.8371 0.609544
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 2.32580 0.0839797
\(768\) 0 0
\(769\) 7.36069 0.265433 0.132717 0.991154i \(-0.457630\pi\)
0.132717 + 0.991154i \(0.457630\pi\)
\(770\) 0 0
\(771\) −15.2351 −0.548680
\(772\) 0 0
\(773\) 10.5548 0.379629 0.189815 0.981820i \(-0.439211\pi\)
0.189815 + 0.981820i \(0.439211\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −6.15676 −0.220872
\(778\) 0 0
\(779\) −2.21008 −0.0791843
\(780\) 0 0
\(781\) 52.0000 1.86071
\(782\) 0 0
\(783\) −2.00000 −0.0714742
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −9.04718 −0.322497 −0.161249 0.986914i \(-0.551552\pi\)
−0.161249 + 0.986914i \(0.551552\pi\)
\(788\) 0 0
\(789\) −29.1773 −1.03874
\(790\) 0 0
\(791\) −13.0205 −0.462956
\(792\) 0 0
\(793\) 5.22529 0.185556
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −16.9627 −0.600848 −0.300424 0.953806i \(-0.597128\pi\)
−0.300424 + 0.953806i \(0.597128\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −15.1773 −0.536263
\(802\) 0 0
\(803\) 4.90271 0.173013
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −6.13009 −0.215790
\(808\) 0 0
\(809\) 15.9877 0.562098 0.281049 0.959693i \(-0.409318\pi\)
0.281049 + 0.959693i \(0.409318\pi\)
\(810\) 0 0
\(811\) −8.39350 −0.294736 −0.147368 0.989082i \(-0.547080\pi\)
−0.147368 + 0.989082i \(0.547080\pi\)
\(812\) 0 0
\(813\) 25.7009 0.901369
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 56.3956 1.97303
\(818\) 0 0
\(819\) 0.340173 0.0118866
\(820\) 0 0
\(821\) 38.0288 1.32721 0.663606 0.748082i \(-0.269025\pi\)
0.663606 + 0.748082i \(0.269025\pi\)
\(822\) 0 0
\(823\) 22.1568 0.772336 0.386168 0.922428i \(-0.373799\pi\)
0.386168 + 0.922428i \(0.373799\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −14.8227 −0.515437 −0.257718 0.966220i \(-0.582971\pi\)
−0.257718 + 0.966220i \(0.582971\pi\)
\(828\) 0 0
\(829\) 13.9467 0.484388 0.242194 0.970228i \(-0.422133\pi\)
0.242194 + 0.970228i \(0.422133\pi\)
\(830\) 0 0
\(831\) −11.2039 −0.388660
\(832\) 0 0
\(833\) 5.75872 0.199528
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −8.34017 −0.288279
\(838\) 0 0
\(839\) −25.4764 −0.879543 −0.439772 0.898110i \(-0.644941\pi\)
−0.439772 + 0.898110i \(0.644941\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) 24.3545 0.838815
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −0.360692 −0.0123935
\(848\) 0 0
\(849\) 6.15676 0.211299
\(850\) 0 0
\(851\) 52.3135 1.79328
\(852\) 0 0
\(853\) 58.1588 1.99132 0.995660 0.0930604i \(-0.0296650\pi\)
0.995660 + 0.0930604i \(0.0296650\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −28.9093 −0.987524 −0.493762 0.869597i \(-0.664379\pi\)
−0.493762 + 0.869597i \(0.664379\pi\)
\(858\) 0 0
\(859\) 50.1834 1.71224 0.856118 0.516780i \(-0.172870\pi\)
0.856118 + 0.516780i \(0.172870\pi\)
\(860\) 0 0
\(861\) 0.340173 0.0115931
\(862\) 0 0
\(863\) 8.13009 0.276752 0.138376 0.990380i \(-0.455812\pi\)
0.138376 + 0.990380i \(0.455812\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 16.1629 0.548921
\(868\) 0 0
\(869\) −28.3135 −0.960470
\(870\) 0 0
\(871\) 5.04718 0.171017
\(872\) 0 0
\(873\) −6.49693 −0.219888
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −26.8371 −0.906225 −0.453112 0.891453i \(-0.649686\pi\)
−0.453112 + 0.891453i \(0.649686\pi\)
\(878\) 0 0
\(879\) −1.75872 −0.0593203
\(880\) 0 0
\(881\) −44.4846 −1.49873 −0.749363 0.662160i \(-0.769640\pi\)
−0.749363 + 0.662160i \(0.769640\pi\)
\(882\) 0 0
\(883\) −20.8781 −0.702605 −0.351303 0.936262i \(-0.614261\pi\)
−0.351303 + 0.936262i \(0.614261\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −22.4079 −0.752383 −0.376191 0.926542i \(-0.622766\pi\)
−0.376191 + 0.926542i \(0.622766\pi\)
\(888\) 0 0
\(889\) −1.84324 −0.0618204
\(890\) 0 0
\(891\) −3.26180 −0.109274
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −2.89043 −0.0965085
\(898\) 0 0
\(899\) 16.6803 0.556321
\(900\) 0 0
\(901\) −48.0288 −1.60007
\(902\) 0 0
\(903\) −8.68035 −0.288864
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −20.8781 −0.693247 −0.346624 0.938004i \(-0.612672\pi\)
−0.346624 + 0.938004i \(0.612672\pi\)
\(908\) 0 0
\(909\) −2.18342 −0.0724194
\(910\) 0 0
\(911\) 3.52198 0.116688 0.0583442 0.998297i \(-0.481418\pi\)
0.0583442 + 0.998297i \(0.481418\pi\)
\(912\) 0 0
\(913\) −22.3012 −0.738063
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −0.313511 −0.0103530
\(918\) 0 0
\(919\) 12.1978 0.402368 0.201184 0.979553i \(-0.435521\pi\)
0.201184 + 0.979553i \(0.435521\pi\)
\(920\) 0 0
\(921\) 27.2039 0.896400
\(922\) 0 0
\(923\) −5.42309 −0.178503
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −5.84324 −0.191917
\(928\) 0 0
\(929\) 2.86376 0.0939570 0.0469785 0.998896i \(-0.485041\pi\)
0.0469785 + 0.998896i \(0.485041\pi\)
\(930\) 0 0
\(931\) −6.49693 −0.212928
\(932\) 0 0
\(933\) −21.8432 −0.715116
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 26.7480 0.873821 0.436910 0.899505i \(-0.356073\pi\)
0.436910 + 0.899505i \(0.356073\pi\)
\(938\) 0 0
\(939\) −25.3874 −0.828485
\(940\) 0 0
\(941\) 10.0144 0.326459 0.163230 0.986588i \(-0.447809\pi\)
0.163230 + 0.986588i \(0.447809\pi\)
\(942\) 0 0
\(943\) −2.89043 −0.0941252
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 47.2183 1.53439 0.767194 0.641415i \(-0.221652\pi\)
0.767194 + 0.641415i \(0.221652\pi\)
\(948\) 0 0
\(949\) −0.511304 −0.0165976
\(950\) 0 0
\(951\) 22.6947 0.735927
\(952\) 0 0
\(953\) 14.6660 0.475077 0.237539 0.971378i \(-0.423659\pi\)
0.237539 + 0.971378i \(0.423659\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 6.52359 0.210878
\(958\) 0 0
\(959\) −2.18342 −0.0705062
\(960\) 0 0
\(961\) 38.5585 1.24382
\(962\) 0 0
\(963\) −8.18342 −0.263707
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −57.3894 −1.84552 −0.922760 0.385375i \(-0.874072\pi\)
−0.922760 + 0.385375i \(0.874072\pi\)
\(968\) 0 0
\(969\) −37.4140 −1.20191
\(970\) 0 0
\(971\) −23.4017 −0.750997 −0.375499 0.926823i \(-0.622529\pi\)
−0.375499 + 0.926823i \(0.622529\pi\)
\(972\) 0 0
\(973\) −1.02052 −0.0327163
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 3.54411 0.113386 0.0566931 0.998392i \(-0.481944\pi\)
0.0566931 + 0.998392i \(0.481944\pi\)
\(978\) 0 0
\(979\) 49.5052 1.58219
\(980\) 0 0
\(981\) 16.8371 0.537567
\(982\) 0 0
\(983\) 11.7152 0.373658 0.186829 0.982392i \(-0.440179\pi\)
0.186829 + 0.982392i \(0.440179\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 73.7563 2.34531
\(990\) 0 0
\(991\) 56.5113 1.79514 0.897570 0.440871i \(-0.145330\pi\)
0.897570 + 0.440871i \(0.145330\pi\)
\(992\) 0 0
\(993\) 5.47641 0.173789
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 40.9048 1.29547 0.647734 0.761867i \(-0.275717\pi\)
0.647734 + 0.761867i \(0.275717\pi\)
\(998\) 0 0
\(999\) −6.15676 −0.194791
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 4200.2.a.bp.1.2 3
4.3 odd 2 8400.2.a.di.1.2 3
5.2 odd 4 840.2.t.d.169.1 6
5.3 odd 4 840.2.t.d.169.4 yes 6
5.4 even 2 4200.2.a.bn.1.2 3
15.2 even 4 2520.2.t.k.1009.6 6
15.8 even 4 2520.2.t.k.1009.5 6
20.3 even 4 1680.2.t.j.1009.1 6
20.7 even 4 1680.2.t.j.1009.4 6
20.19 odd 2 8400.2.a.dl.1.2 3
60.23 odd 4 5040.2.t.z.1009.5 6
60.47 odd 4 5040.2.t.z.1009.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
840.2.t.d.169.1 6 5.2 odd 4
840.2.t.d.169.4 yes 6 5.3 odd 4
1680.2.t.j.1009.1 6 20.3 even 4
1680.2.t.j.1009.4 6 20.7 even 4
2520.2.t.k.1009.5 6 15.8 even 4
2520.2.t.k.1009.6 6 15.2 even 4
4200.2.a.bn.1.2 3 5.4 even 2
4200.2.a.bp.1.2 3 1.1 even 1 trivial
5040.2.t.z.1009.5 6 60.23 odd 4
5040.2.t.z.1009.6 6 60.47 odd 4
8400.2.a.di.1.2 3 4.3 odd 2
8400.2.a.dl.1.2 3 20.19 odd 2