Properties

Label 1900.2.a.g.1.1
Level $1900$
Weight $2$
Character 1900.1
Self dual yes
Analytic conductor $15.172$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1900.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(15.1715763840\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.321.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.69963\) of defining polynomial
Character \(\chi\) \(=\) 1900.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.58836 q^{3} -2.81089 q^{7} +3.69963 q^{9} +O(q^{10})\) \(q-2.58836 q^{3} -2.81089 q^{7} +3.69963 q^{9} +1.58836 q^{11} -0.888736 q^{13} +3.98762 q^{17} +1.00000 q^{19} +7.27561 q^{21} -3.30037 q^{23} -1.81089 q^{27} +6.98762 q^{29} -4.51052 q^{31} -4.11126 q^{33} +2.41164 q^{37} +2.30037 q^{39} +5.09888 q^{41} -8.17673 q^{43} +9.08650 q^{47} +0.901116 q^{49} -10.3214 q^{51} -7.79851 q^{53} -2.58836 q^{57} -2.22253 q^{59} -9.90978 q^{61} -10.3993 q^{63} +7.56360 q^{67} +8.54256 q^{69} -0.777472 q^{71} +0.876356 q^{73} -4.46472 q^{77} -8.94182 q^{79} -6.41164 q^{81} -11.6996 q^{83} -18.0865 q^{87} -5.27561 q^{89} +2.49814 q^{91} +11.6749 q^{93} -7.24219 q^{97} +5.87636 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{3} - 2 q^{7} + 5 q^{9} + O(q^{10}) \) \( 3 q - 2 q^{3} - 2 q^{7} + 5 q^{9} - q^{11} - 3 q^{13} - 6 q^{17} + 3 q^{19} - 8 q^{21} - 16 q^{23} + q^{27} + 3 q^{29} - q^{31} - 12 q^{33} + 13 q^{37} + 13 q^{39} - 3 q^{41} - 13 q^{43} - 9 q^{47} + 21 q^{49} - 12 q^{51} + q^{53} - 2 q^{57} - 6 q^{59} - 5 q^{61} - 19 q^{63} - 19 q^{67} + 9 q^{69} - 3 q^{71} - 15 q^{73} + 10 q^{77} + 2 q^{79} - 25 q^{81} - 29 q^{83} - 18 q^{87} + 14 q^{89} - 23 q^{91} - 7 q^{93} + q^{97} + O(q^{100}) \)

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
\(3\) −2.58836 −1.49439 −0.747196 0.664603i \(-0.768601\pi\)
−0.747196 + 0.664603i \(0.768601\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −2.81089 −1.06242 −0.531209 0.847241i \(-0.678262\pi\)
−0.531209 + 0.847241i \(0.678262\pi\)
\(8\) 0 0
\(9\) 3.69963 1.23321
\(10\) 0 0
\(11\) 1.58836 0.478910 0.239455 0.970907i \(-0.423031\pi\)
0.239455 + 0.970907i \(0.423031\pi\)
\(12\) 0 0
\(13\) −0.888736 −0.246491 −0.123245 0.992376i \(-0.539330\pi\)
−0.123245 + 0.992376i \(0.539330\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.98762 0.967140 0.483570 0.875306i \(-0.339340\pi\)
0.483570 + 0.875306i \(0.339340\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 7.27561 1.58767
\(22\) 0 0
\(23\) −3.30037 −0.688175 −0.344088 0.938938i \(-0.611812\pi\)
−0.344088 + 0.938938i \(0.611812\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −1.81089 −0.348506
\(28\) 0 0
\(29\) 6.98762 1.29757 0.648784 0.760972i \(-0.275278\pi\)
0.648784 + 0.760972i \(0.275278\pi\)
\(30\) 0 0
\(31\) −4.51052 −0.810113 −0.405057 0.914292i \(-0.632748\pi\)
−0.405057 + 0.914292i \(0.632748\pi\)
\(32\) 0 0
\(33\) −4.11126 −0.715679
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.41164 0.396471 0.198235 0.980154i \(-0.436479\pi\)
0.198235 + 0.980154i \(0.436479\pi\)
\(38\) 0 0
\(39\) 2.30037 0.368354
\(40\) 0 0
\(41\) 5.09888 0.796312 0.398156 0.917318i \(-0.369650\pi\)
0.398156 + 0.917318i \(0.369650\pi\)
\(42\) 0 0
\(43\) −8.17673 −1.24694 −0.623470 0.781848i \(-0.714278\pi\)
−0.623470 + 0.781848i \(0.714278\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.08650 1.32540 0.662701 0.748884i \(-0.269410\pi\)
0.662701 + 0.748884i \(0.269410\pi\)
\(48\) 0 0
\(49\) 0.901116 0.128731
\(50\) 0 0
\(51\) −10.3214 −1.44529
\(52\) 0 0
\(53\) −7.79851 −1.07121 −0.535604 0.844469i \(-0.679916\pi\)
−0.535604 + 0.844469i \(0.679916\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −2.58836 −0.342837
\(58\) 0 0
\(59\) −2.22253 −0.289349 −0.144674 0.989479i \(-0.546213\pi\)
−0.144674 + 0.989479i \(0.546213\pi\)
\(60\) 0 0
\(61\) −9.90978 −1.26882 −0.634408 0.772998i \(-0.718756\pi\)
−0.634408 + 0.772998i \(0.718756\pi\)
\(62\) 0 0
\(63\) −10.3993 −1.31018
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 7.56360 0.924041 0.462021 0.886869i \(-0.347125\pi\)
0.462021 + 0.886869i \(0.347125\pi\)
\(68\) 0 0
\(69\) 8.54256 1.02840
\(70\) 0 0
\(71\) −0.777472 −0.0922689 −0.0461345 0.998935i \(-0.514690\pi\)
−0.0461345 + 0.998935i \(0.514690\pi\)
\(72\) 0 0
\(73\) 0.876356 0.102570 0.0512849 0.998684i \(-0.483668\pi\)
0.0512849 + 0.998684i \(0.483668\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4.46472 −0.508802
\(78\) 0 0
\(79\) −8.94182 −1.00603 −0.503017 0.864277i \(-0.667777\pi\)
−0.503017 + 0.864277i \(0.667777\pi\)
\(80\) 0 0
\(81\) −6.41164 −0.712404
\(82\) 0 0
\(83\) −11.6996 −1.28420 −0.642101 0.766620i \(-0.721937\pi\)
−0.642101 + 0.766620i \(0.721937\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −18.0865 −1.93908
\(88\) 0 0
\(89\) −5.27561 −0.559214 −0.279607 0.960115i \(-0.590204\pi\)
−0.279607 + 0.960115i \(0.590204\pi\)
\(90\) 0 0
\(91\) 2.49814 0.261876
\(92\) 0 0
\(93\) 11.6749 1.21063
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −7.24219 −0.735333 −0.367667 0.929958i \(-0.619843\pi\)
−0.367667 + 0.929958i \(0.619843\pi\)
\(98\) 0 0
\(99\) 5.87636 0.590596
\(100\) 0 0
\(101\) −0.143307 −0.0142596 −0.00712981 0.999975i \(-0.502270\pi\)
−0.00712981 + 0.999975i \(0.502270\pi\)
\(102\) 0 0
\(103\) 11.8319 1.16584 0.582918 0.812531i \(-0.301911\pi\)
0.582918 + 0.812531i \(0.301911\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −10.9222 −1.05588 −0.527942 0.849280i \(-0.677036\pi\)
−0.527942 + 0.849280i \(0.677036\pi\)
\(108\) 0 0
\(109\) 15.6749 1.50138 0.750690 0.660655i \(-0.229721\pi\)
0.750690 + 0.660655i \(0.229721\pi\)
\(110\) 0 0
\(111\) −6.24219 −0.592483
\(112\) 0 0
\(113\) −6.33379 −0.595833 −0.297917 0.954592i \(-0.596292\pi\)
−0.297917 + 0.954592i \(0.596292\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −3.28799 −0.303975
\(118\) 0 0
\(119\) −11.2088 −1.02751
\(120\) 0 0
\(121\) −8.47710 −0.770645
\(122\) 0 0
\(123\) −13.1978 −1.19000
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −11.7207 −1.04004 −0.520021 0.854154i \(-0.674076\pi\)
−0.520021 + 0.854154i \(0.674076\pi\)
\(128\) 0 0
\(129\) 21.1643 1.86342
\(130\) 0 0
\(131\) −5.45234 −0.476373 −0.238187 0.971219i \(-0.576553\pi\)
−0.238187 + 0.971219i \(0.576553\pi\)
\(132\) 0 0
\(133\) −2.81089 −0.243735
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −14.8764 −1.27097 −0.635486 0.772112i \(-0.719200\pi\)
−0.635486 + 0.772112i \(0.719200\pi\)
\(138\) 0 0
\(139\) −21.9294 −1.86003 −0.930015 0.367520i \(-0.880207\pi\)
−0.930015 + 0.367520i \(0.880207\pi\)
\(140\) 0 0
\(141\) −23.5192 −1.98067
\(142\) 0 0
\(143\) −1.41164 −0.118047
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −2.33242 −0.192374
\(148\) 0 0
\(149\) 10.1447 0.831085 0.415542 0.909574i \(-0.363592\pi\)
0.415542 + 0.909574i \(0.363592\pi\)
\(150\) 0 0
\(151\) −9.66621 −0.786625 −0.393312 0.919405i \(-0.628671\pi\)
−0.393312 + 0.919405i \(0.628671\pi\)
\(152\) 0 0
\(153\) 14.7527 1.19269
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 8.25457 0.658787 0.329393 0.944193i \(-0.393156\pi\)
0.329393 + 0.944193i \(0.393156\pi\)
\(158\) 0 0
\(159\) 20.1854 1.60081
\(160\) 0 0
\(161\) 9.27699 0.731129
\(162\) 0 0
\(163\) −9.46472 −0.741334 −0.370667 0.928766i \(-0.620871\pi\)
−0.370667 + 0.928766i \(0.620871\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.92216 0.148741 0.0743705 0.997231i \(-0.476305\pi\)
0.0743705 + 0.997231i \(0.476305\pi\)
\(168\) 0 0
\(169\) −12.2101 −0.939242
\(170\) 0 0
\(171\) 3.69963 0.282918
\(172\) 0 0
\(173\) 22.3411 1.69856 0.849280 0.527942i \(-0.177036\pi\)
0.849280 + 0.527942i \(0.177036\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 5.75271 0.432400
\(178\) 0 0
\(179\) −21.7738 −1.62745 −0.813723 0.581252i \(-0.802563\pi\)
−0.813723 + 0.581252i \(0.802563\pi\)
\(180\) 0 0
\(181\) −7.00000 −0.520306 −0.260153 0.965567i \(-0.583773\pi\)
−0.260153 + 0.965567i \(0.583773\pi\)
\(182\) 0 0
\(183\) 25.6501 1.89611
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 6.33379 0.463173
\(188\) 0 0
\(189\) 5.09022 0.370259
\(190\) 0 0
\(191\) 3.21015 0.232278 0.116139 0.993233i \(-0.462948\pi\)
0.116139 + 0.993233i \(0.462948\pi\)
\(192\) 0 0
\(193\) 0.522900 0.0376392 0.0188196 0.999823i \(-0.494009\pi\)
0.0188196 + 0.999823i \(0.494009\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.60940 0.185912 0.0929562 0.995670i \(-0.470368\pi\)
0.0929562 + 0.995670i \(0.470368\pi\)
\(198\) 0 0
\(199\) −12.8960 −0.914175 −0.457087 0.889422i \(-0.651107\pi\)
−0.457087 + 0.889422i \(0.651107\pi\)
\(200\) 0 0
\(201\) −19.5774 −1.38088
\(202\) 0 0
\(203\) −19.6414 −1.37856
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −12.2101 −0.848664
\(208\) 0 0
\(209\) 1.58836 0.109869
\(210\) 0 0
\(211\) 23.3535 1.60772 0.803859 0.594820i \(-0.202777\pi\)
0.803859 + 0.594820i \(0.202777\pi\)
\(212\) 0 0
\(213\) 2.01238 0.137886
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 12.6786 0.860679
\(218\) 0 0
\(219\) −2.26833 −0.153279
\(220\) 0 0
\(221\) −3.54394 −0.238391
\(222\) 0 0
\(223\) 3.50914 0.234990 0.117495 0.993073i \(-0.462514\pi\)
0.117495 + 0.993073i \(0.462514\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 12.0531 0.799991 0.399996 0.916517i \(-0.369012\pi\)
0.399996 + 0.916517i \(0.369012\pi\)
\(228\) 0 0
\(229\) 21.0407 1.39041 0.695204 0.718812i \(-0.255314\pi\)
0.695204 + 0.718812i \(0.255314\pi\)
\(230\) 0 0
\(231\) 11.5563 0.760350
\(232\) 0 0
\(233\) −23.4720 −1.53770 −0.768851 0.639428i \(-0.779171\pi\)
−0.768851 + 0.639428i \(0.779171\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 23.1447 1.50341
\(238\) 0 0
\(239\) 0.263233 0.0170271 0.00851355 0.999964i \(-0.497290\pi\)
0.00851355 + 0.999964i \(0.497290\pi\)
\(240\) 0 0
\(241\) −19.8974 −1.28170 −0.640852 0.767664i \(-0.721419\pi\)
−0.640852 + 0.767664i \(0.721419\pi\)
\(242\) 0 0
\(243\) 22.0283 1.41312
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −0.888736 −0.0565489
\(248\) 0 0
\(249\) 30.2829 1.91910
\(250\) 0 0
\(251\) 15.1964 0.959188 0.479594 0.877491i \(-0.340784\pi\)
0.479594 + 0.877491i \(0.340784\pi\)
\(252\) 0 0
\(253\) −5.24219 −0.329574
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −21.8072 −1.36029 −0.680147 0.733076i \(-0.738084\pi\)
−0.680147 + 0.733076i \(0.738084\pi\)
\(258\) 0 0
\(259\) −6.77885 −0.421217
\(260\) 0 0
\(261\) 25.8516 1.60017
\(262\) 0 0
\(263\) −19.8850 −1.22616 −0.613081 0.790020i \(-0.710070\pi\)
−0.613081 + 0.790020i \(0.710070\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 13.6552 0.835685
\(268\) 0 0
\(269\) −31.2880 −1.90766 −0.953831 0.300343i \(-0.902899\pi\)
−0.953831 + 0.300343i \(0.902899\pi\)
\(270\) 0 0
\(271\) −11.3411 −0.688921 −0.344461 0.938801i \(-0.611938\pi\)
−0.344461 + 0.938801i \(0.611938\pi\)
\(272\) 0 0
\(273\) −6.46610 −0.391346
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 18.1520 1.09065 0.545323 0.838226i \(-0.316407\pi\)
0.545323 + 0.838226i \(0.316407\pi\)
\(278\) 0 0
\(279\) −16.6872 −0.999039
\(280\) 0 0
\(281\) −6.42030 −0.383003 −0.191501 0.981492i \(-0.561336\pi\)
−0.191501 + 0.981492i \(0.561336\pi\)
\(282\) 0 0
\(283\) −19.2051 −1.14162 −0.570811 0.821082i \(-0.693371\pi\)
−0.570811 + 0.821082i \(0.693371\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −14.3324 −0.846016
\(288\) 0 0
\(289\) −1.09888 −0.0646403
\(290\) 0 0
\(291\) 18.7454 1.09888
\(292\) 0 0
\(293\) 32.5192 1.89979 0.949895 0.312568i \(-0.101189\pi\)
0.949895 + 0.312568i \(0.101189\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −2.87636 −0.166903
\(298\) 0 0
\(299\) 2.93316 0.169629
\(300\) 0 0
\(301\) 22.9839 1.32477
\(302\) 0 0
\(303\) 0.370932 0.0213095
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 26.7810 1.52847 0.764237 0.644935i \(-0.223116\pi\)
0.764237 + 0.644935i \(0.223116\pi\)
\(308\) 0 0
\(309\) −30.6253 −1.74222
\(310\) 0 0
\(311\) 22.2880 1.26384 0.631918 0.775035i \(-0.282268\pi\)
0.631918 + 0.775035i \(0.282268\pi\)
\(312\) 0 0
\(313\) −22.9629 −1.29794 −0.648969 0.760815i \(-0.724799\pi\)
−0.648969 + 0.760815i \(0.724799\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 28.1840 1.58297 0.791486 0.611187i \(-0.209308\pi\)
0.791486 + 0.611187i \(0.209308\pi\)
\(318\) 0 0
\(319\) 11.0989 0.621418
\(320\) 0 0
\(321\) 28.2705 1.57791
\(322\) 0 0
\(323\) 3.98762 0.221877
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −40.5723 −2.24365
\(328\) 0 0
\(329\) −25.5412 −1.40813
\(330\) 0 0
\(331\) 19.1716 1.05377 0.526884 0.849937i \(-0.323360\pi\)
0.526884 + 0.849937i \(0.323360\pi\)
\(332\) 0 0
\(333\) 8.92216 0.488931
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −14.2632 −0.776968 −0.388484 0.921456i \(-0.627001\pi\)
−0.388484 + 0.921456i \(0.627001\pi\)
\(338\) 0 0
\(339\) 16.3942 0.890409
\(340\) 0 0
\(341\) −7.16435 −0.387971
\(342\) 0 0
\(343\) 17.1433 0.925652
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −1.90249 −0.102131 −0.0510656 0.998695i \(-0.516262\pi\)
−0.0510656 + 0.998695i \(0.516262\pi\)
\(348\) 0 0
\(349\) −0.556321 −0.0297792 −0.0148896 0.999889i \(-0.504740\pi\)
−0.0148896 + 0.999889i \(0.504740\pi\)
\(350\) 0 0
\(351\) 1.60940 0.0859037
\(352\) 0 0
\(353\) 13.1075 0.697644 0.348822 0.937189i \(-0.386582\pi\)
0.348822 + 0.937189i \(0.386582\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 29.0124 1.53550
\(358\) 0 0
\(359\) −21.0197 −1.10938 −0.554688 0.832059i \(-0.687162\pi\)
−0.554688 + 0.832059i \(0.687162\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 21.9418 1.15165
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 30.8021 1.60786 0.803928 0.594727i \(-0.202740\pi\)
0.803928 + 0.594727i \(0.202740\pi\)
\(368\) 0 0
\(369\) 18.8640 0.982019
\(370\) 0 0
\(371\) 21.9208 1.13807
\(372\) 0 0
\(373\) 10.0902 0.522452 0.261226 0.965278i \(-0.415873\pi\)
0.261226 + 0.965278i \(0.415873\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −6.21015 −0.319839
\(378\) 0 0
\(379\) 31.1286 1.59897 0.799484 0.600687i \(-0.205106\pi\)
0.799484 + 0.600687i \(0.205106\pi\)
\(380\) 0 0
\(381\) 30.3374 1.55423
\(382\) 0 0
\(383\) −11.4895 −0.587085 −0.293542 0.955946i \(-0.594834\pi\)
−0.293542 + 0.955946i \(0.594834\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −30.2509 −1.53774
\(388\) 0 0
\(389\) −35.0146 −1.77531 −0.887655 0.460510i \(-0.847667\pi\)
−0.887655 + 0.460510i \(0.847667\pi\)
\(390\) 0 0
\(391\) −13.1606 −0.665562
\(392\) 0 0
\(393\) 14.1126 0.711889
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 5.88874 0.295547 0.147774 0.989021i \(-0.452789\pi\)
0.147774 + 0.989021i \(0.452789\pi\)
\(398\) 0 0
\(399\) 7.27561 0.364236
\(400\) 0 0
\(401\) 4.23491 0.211481 0.105741 0.994394i \(-0.466279\pi\)
0.105741 + 0.994394i \(0.466279\pi\)
\(402\) 0 0
\(403\) 4.00866 0.199686
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 3.83056 0.189874
\(408\) 0 0
\(409\) −7.05308 −0.348753 −0.174376 0.984679i \(-0.555791\pi\)
−0.174376 + 0.984679i \(0.555791\pi\)
\(410\) 0 0
\(411\) 38.5054 1.89933
\(412\) 0 0
\(413\) 6.24729 0.307409
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 56.7614 2.77962
\(418\) 0 0
\(419\) 17.7724 0.868237 0.434119 0.900856i \(-0.357060\pi\)
0.434119 + 0.900856i \(0.357060\pi\)
\(420\) 0 0
\(421\) −5.81818 −0.283561 −0.141780 0.989898i \(-0.545283\pi\)
−0.141780 + 0.989898i \(0.545283\pi\)
\(422\) 0 0
\(423\) 33.6167 1.63450
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 27.8553 1.34801
\(428\) 0 0
\(429\) 3.65383 0.176408
\(430\) 0 0
\(431\) −23.1854 −1.11680 −0.558400 0.829572i \(-0.688585\pi\)
−0.558400 + 0.829572i \(0.688585\pi\)
\(432\) 0 0
\(433\) 12.3462 0.593319 0.296660 0.954983i \(-0.404127\pi\)
0.296660 + 0.954983i \(0.404127\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −3.30037 −0.157878
\(438\) 0 0
\(439\) 23.8502 1.13831 0.569154 0.822231i \(-0.307271\pi\)
0.569154 + 0.822231i \(0.307271\pi\)
\(440\) 0 0
\(441\) 3.33379 0.158752
\(442\) 0 0
\(443\) −32.6291 −1.55025 −0.775127 0.631806i \(-0.782314\pi\)
−0.775127 + 0.631806i \(0.782314\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −26.2581 −1.24197
\(448\) 0 0
\(449\) 29.4152 1.38819 0.694095 0.719884i \(-0.255805\pi\)
0.694095 + 0.719884i \(0.255805\pi\)
\(450\) 0 0
\(451\) 8.09888 0.381362
\(452\) 0 0
\(453\) 25.0197 1.17553
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 26.2880 1.22970 0.614850 0.788644i \(-0.289216\pi\)
0.614850 + 0.788644i \(0.289216\pi\)
\(458\) 0 0
\(459\) −7.22115 −0.337054
\(460\) 0 0
\(461\) −12.5068 −0.582500 −0.291250 0.956647i \(-0.594071\pi\)
−0.291250 + 0.956647i \(0.594071\pi\)
\(462\) 0 0
\(463\) 23.7083 1.10182 0.550909 0.834565i \(-0.314281\pi\)
0.550909 + 0.834565i \(0.314281\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −30.4734 −1.41014 −0.705070 0.709138i \(-0.749084\pi\)
−0.705070 + 0.709138i \(0.749084\pi\)
\(468\) 0 0
\(469\) −21.2605 −0.981718
\(470\) 0 0
\(471\) −21.3658 −0.984486
\(472\) 0 0
\(473\) −12.9876 −0.597171
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −28.8516 −1.32102
\(478\) 0 0
\(479\) −12.6676 −0.578797 −0.289398 0.957209i \(-0.593455\pi\)
−0.289398 + 0.957209i \(0.593455\pi\)
\(480\) 0 0
\(481\) −2.14331 −0.0977264
\(482\) 0 0
\(483\) −24.0122 −1.09259
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −43.2051 −1.95781 −0.978904 0.204321i \(-0.934501\pi\)
−0.978904 + 0.204321i \(0.934501\pi\)
\(488\) 0 0
\(489\) 24.4981 1.10784
\(490\) 0 0
\(491\) 18.4065 0.830676 0.415338 0.909667i \(-0.363663\pi\)
0.415338 + 0.909667i \(0.363663\pi\)
\(492\) 0 0
\(493\) 27.8640 1.25493
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2.18539 0.0980281
\(498\) 0 0
\(499\) −13.9208 −0.623180 −0.311590 0.950217i \(-0.600861\pi\)
−0.311590 + 0.950217i \(0.600861\pi\)
\(500\) 0 0
\(501\) −4.97524 −0.222277
\(502\) 0 0
\(503\) 18.8777 0.841717 0.420858 0.907126i \(-0.361729\pi\)
0.420858 + 0.907126i \(0.361729\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 31.6043 1.40360
\(508\) 0 0
\(509\) −7.63554 −0.338439 −0.169220 0.985578i \(-0.554125\pi\)
−0.169220 + 0.985578i \(0.554125\pi\)
\(510\) 0 0
\(511\) −2.46334 −0.108972
\(512\) 0 0
\(513\) −1.81089 −0.0799528
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 14.4327 0.634748
\(518\) 0 0
\(519\) −57.8268 −2.53832
\(520\) 0 0
\(521\) −23.3324 −1.02221 −0.511106 0.859518i \(-0.670764\pi\)
−0.511106 + 0.859518i \(0.670764\pi\)
\(522\) 0 0
\(523\) 13.6735 0.597900 0.298950 0.954269i \(-0.403364\pi\)
0.298950 + 0.954269i \(0.403364\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −17.9862 −0.783493
\(528\) 0 0
\(529\) −12.1075 −0.526415
\(530\) 0 0
\(531\) −8.22253 −0.356827
\(532\) 0 0
\(533\) −4.53156 −0.196284
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 56.3584 2.43204
\(538\) 0 0
\(539\) 1.43130 0.0616504
\(540\) 0 0
\(541\) −35.4472 −1.52400 −0.761998 0.647579i \(-0.775781\pi\)
−0.761998 + 0.647579i \(0.775781\pi\)
\(542\) 0 0
\(543\) 18.1185 0.777541
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −13.8392 −0.591722 −0.295861 0.955231i \(-0.595607\pi\)
−0.295861 + 0.955231i \(0.595607\pi\)
\(548\) 0 0
\(549\) −36.6625 −1.56472
\(550\) 0 0
\(551\) 6.98762 0.297683
\(552\) 0 0
\(553\) 25.1345 1.06883
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −30.8777 −1.30833 −0.654166 0.756351i \(-0.726980\pi\)
−0.654166 + 0.756351i \(0.726980\pi\)
\(558\) 0 0
\(559\) 7.26695 0.307359
\(560\) 0 0
\(561\) −16.3942 −0.692162
\(562\) 0 0
\(563\) 6.38688 0.269175 0.134587 0.990902i \(-0.457029\pi\)
0.134587 + 0.990902i \(0.457029\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 18.0224 0.756870
\(568\) 0 0
\(569\) −43.2334 −1.81244 −0.906219 0.422809i \(-0.861044\pi\)
−0.906219 + 0.422809i \(0.861044\pi\)
\(570\) 0 0
\(571\) −33.8726 −1.41753 −0.708763 0.705447i \(-0.750746\pi\)
−0.708763 + 0.705447i \(0.750746\pi\)
\(572\) 0 0
\(573\) −8.30903 −0.347115
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −15.0975 −0.628517 −0.314259 0.949337i \(-0.601756\pi\)
−0.314259 + 0.949337i \(0.601756\pi\)
\(578\) 0 0
\(579\) −1.35346 −0.0562477
\(580\) 0 0
\(581\) 32.8864 1.36436
\(582\) 0 0
\(583\) −12.3869 −0.513012
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −40.7651 −1.68256 −0.841278 0.540603i \(-0.818196\pi\)
−0.841278 + 0.540603i \(0.818196\pi\)
\(588\) 0 0
\(589\) −4.51052 −0.185853
\(590\) 0 0
\(591\) −6.75409 −0.277826
\(592\) 0 0
\(593\) 5.77609 0.237196 0.118598 0.992942i \(-0.462160\pi\)
0.118598 + 0.992942i \(0.462160\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 33.3796 1.36614
\(598\) 0 0
\(599\) −14.0087 −0.572378 −0.286189 0.958173i \(-0.592389\pi\)
−0.286189 + 0.958173i \(0.592389\pi\)
\(600\) 0 0
\(601\) −23.9222 −0.975805 −0.487903 0.872898i \(-0.662238\pi\)
−0.487903 + 0.872898i \(0.662238\pi\)
\(602\) 0 0
\(603\) 27.9825 1.13954
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 18.1026 0.734762 0.367381 0.930071i \(-0.380255\pi\)
0.367381 + 0.930071i \(0.380255\pi\)
\(608\) 0 0
\(609\) 50.8392 2.06011
\(610\) 0 0
\(611\) −8.07550 −0.326700
\(612\) 0 0
\(613\) −41.9701 −1.69516 −0.847579 0.530669i \(-0.821941\pi\)
−0.847579 + 0.530669i \(0.821941\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −15.9112 −0.640559 −0.320279 0.947323i \(-0.603777\pi\)
−0.320279 + 0.947323i \(0.603777\pi\)
\(618\) 0 0
\(619\) −27.8726 −1.12030 −0.560148 0.828393i \(-0.689256\pi\)
−0.560148 + 0.828393i \(0.689256\pi\)
\(620\) 0 0
\(621\) 5.97662 0.239833
\(622\) 0 0
\(623\) 14.8292 0.594118
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −4.11126 −0.164188
\(628\) 0 0
\(629\) 9.61669 0.383442
\(630\) 0 0
\(631\) 33.3214 1.32650 0.663252 0.748396i \(-0.269176\pi\)
0.663252 + 0.748396i \(0.269176\pi\)
\(632\) 0 0
\(633\) −60.4472 −2.40256
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −0.800854 −0.0317310
\(638\) 0 0
\(639\) −2.87636 −0.113787
\(640\) 0 0
\(641\) 19.7985 0.781994 0.390997 0.920392i \(-0.372130\pi\)
0.390997 + 0.920392i \(0.372130\pi\)
\(642\) 0 0
\(643\) 22.0000 0.867595 0.433798 0.901010i \(-0.357173\pi\)
0.433798 + 0.901010i \(0.357173\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −29.3818 −1.15512 −0.577558 0.816349i \(-0.695994\pi\)
−0.577558 + 0.816349i \(0.695994\pi\)
\(648\) 0 0
\(649\) −3.53018 −0.138572
\(650\) 0 0
\(651\) −32.8168 −1.28619
\(652\) 0 0
\(653\) 18.8306 0.736897 0.368448 0.929648i \(-0.379889\pi\)
0.368448 + 0.929648i \(0.379889\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 3.24219 0.126490
\(658\) 0 0
\(659\) −44.7293 −1.74241 −0.871204 0.490922i \(-0.836660\pi\)
−0.871204 + 0.490922i \(0.836660\pi\)
\(660\) 0 0
\(661\) 3.28799 0.127888 0.0639440 0.997953i \(-0.479632\pi\)
0.0639440 + 0.997953i \(0.479632\pi\)
\(662\) 0 0
\(663\) 9.17301 0.356250
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −23.0617 −0.892954
\(668\) 0 0
\(669\) −9.08294 −0.351167
\(670\) 0 0
\(671\) −15.7403 −0.607649
\(672\) 0 0
\(673\) −1.83703 −0.0708123 −0.0354061 0.999373i \(-0.511272\pi\)
−0.0354061 + 0.999373i \(0.511272\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −26.2953 −1.01061 −0.505305 0.862941i \(-0.668620\pi\)
−0.505305 + 0.862941i \(0.668620\pi\)
\(678\) 0 0
\(679\) 20.3570 0.781231
\(680\) 0 0
\(681\) −31.1978 −1.19550
\(682\) 0 0
\(683\) −8.29899 −0.317552 −0.158776 0.987315i \(-0.550755\pi\)
−0.158776 + 0.987315i \(0.550755\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −54.4610 −2.07782
\(688\) 0 0
\(689\) 6.93082 0.264043
\(690\) 0 0
\(691\) 38.2123 1.45367 0.726833 0.686814i \(-0.240991\pi\)
0.726833 + 0.686814i \(0.240991\pi\)
\(692\) 0 0
\(693\) −16.5178 −0.627459
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 20.3324 0.770145
\(698\) 0 0
\(699\) 60.7541 2.29793
\(700\) 0 0
\(701\) 13.2212 0.499356 0.249678 0.968329i \(-0.419675\pi\)
0.249678 + 0.968329i \(0.419675\pi\)
\(702\) 0 0
\(703\) 2.41164 0.0909566
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0.402822 0.0151497
\(708\) 0 0
\(709\) −43.7266 −1.64219 −0.821093 0.570794i \(-0.806635\pi\)
−0.821093 + 0.570794i \(0.806635\pi\)
\(710\) 0 0
\(711\) −33.0814 −1.24065
\(712\) 0 0
\(713\) 14.8864 0.557500
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −0.681342 −0.0254452
\(718\) 0 0
\(719\) 32.1716 1.19980 0.599900 0.800075i \(-0.295207\pi\)
0.599900 + 0.800075i \(0.295207\pi\)
\(720\) 0 0
\(721\) −33.2583 −1.23860
\(722\) 0 0
\(723\) 51.5017 1.91537
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 6.25967 0.232158 0.116079 0.993240i \(-0.462967\pi\)
0.116079 + 0.993240i \(0.462967\pi\)
\(728\) 0 0
\(729\) −37.7824 −1.39935
\(730\) 0 0
\(731\) −32.6057 −1.20596
\(732\) 0 0
\(733\) −4.81955 −0.178014 −0.0890071 0.996031i \(-0.528369\pi\)
−0.0890071 + 0.996031i \(0.528369\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 12.0138 0.442532
\(738\) 0 0
\(739\) 29.3969 1.08138 0.540692 0.841221i \(-0.318163\pi\)
0.540692 + 0.841221i \(0.318163\pi\)
\(740\) 0 0
\(741\) 2.30037 0.0845063
\(742\) 0 0
\(743\) −16.9890 −0.623266 −0.311633 0.950203i \(-0.600876\pi\)
−0.311633 + 0.950203i \(0.600876\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −43.2843 −1.58369
\(748\) 0 0
\(749\) 30.7010 1.12179
\(750\) 0 0
\(751\) 7.98252 0.291286 0.145643 0.989337i \(-0.453475\pi\)
0.145643 + 0.989337i \(0.453475\pi\)
\(752\) 0 0
\(753\) −39.3338 −1.43340
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 14.6749 0.533367 0.266684 0.963784i \(-0.414072\pi\)
0.266684 + 0.963784i \(0.414072\pi\)
\(758\) 0 0
\(759\) 13.5687 0.492513
\(760\) 0 0
\(761\) −9.85807 −0.357355 −0.178677 0.983908i \(-0.557182\pi\)
−0.178677 + 0.983908i \(0.557182\pi\)
\(762\) 0 0
\(763\) −44.0604 −1.59509
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1.97524 0.0713218
\(768\) 0 0
\(769\) −17.7948 −0.641697 −0.320848 0.947131i \(-0.603968\pi\)
−0.320848 + 0.947131i \(0.603968\pi\)
\(770\) 0 0
\(771\) 56.4449 2.03281
\(772\) 0 0
\(773\) 2.34617 0.0843859 0.0421930 0.999109i \(-0.486566\pi\)
0.0421930 + 0.999109i \(0.486566\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 17.5461 0.629464
\(778\) 0 0
\(779\) 5.09888 0.182686
\(780\) 0 0
\(781\) −1.23491 −0.0441885
\(782\) 0 0
\(783\) −12.6538 −0.452211
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 27.2422 0.971079 0.485540 0.874215i \(-0.338623\pi\)
0.485540 + 0.874215i \(0.338623\pi\)
\(788\) 0 0
\(789\) 51.4697 1.83237
\(790\) 0 0
\(791\) 17.8036 0.633023
\(792\) 0 0
\(793\) 8.80717 0.312752
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −12.2261 −0.433070 −0.216535 0.976275i \(-0.569476\pi\)
−0.216535 + 0.976275i \(0.569476\pi\)
\(798\) 0 0
\(799\) 36.2335 1.28185
\(800\) 0 0
\(801\) −19.5178 −0.689628
\(802\) 0 0
\(803\) 1.39197 0.0491216
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 80.9847 2.85080
\(808\) 0 0
\(809\) 9.40063 0.330509 0.165254 0.986251i \(-0.447156\pi\)
0.165254 + 0.986251i \(0.447156\pi\)
\(810\) 0 0
\(811\) 37.9729 1.33341 0.666704 0.745322i \(-0.267704\pi\)
0.666704 + 0.745322i \(0.267704\pi\)
\(812\) 0 0
\(813\) 29.3548 1.02952
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −8.17673 −0.286068
\(818\) 0 0
\(819\) 9.24219 0.322948
\(820\) 0 0
\(821\) −14.0283 −0.489592 −0.244796 0.969575i \(-0.578721\pi\)
−0.244796 + 0.969575i \(0.578721\pi\)
\(822\) 0 0
\(823\) −7.66621 −0.267227 −0.133614 0.991034i \(-0.542658\pi\)
−0.133614 + 0.991034i \(0.542658\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 24.0122 0.834987 0.417493 0.908680i \(-0.362909\pi\)
0.417493 + 0.908680i \(0.362909\pi\)
\(828\) 0 0
\(829\) 19.7527 0.686040 0.343020 0.939328i \(-0.388550\pi\)
0.343020 + 0.939328i \(0.388550\pi\)
\(830\) 0 0
\(831\) −46.9839 −1.62985
\(832\) 0 0
\(833\) 3.59331 0.124501
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 8.16807 0.282330
\(838\) 0 0
\(839\) −27.5736 −0.951948 −0.475974 0.879459i \(-0.657904\pi\)
−0.475974 + 0.879459i \(0.657904\pi\)
\(840\) 0 0
\(841\) 19.8268 0.683684
\(842\) 0 0
\(843\) 16.6181 0.572357
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 23.8282 0.818747
\(848\) 0 0
\(849\) 49.7097 1.70603
\(850\) 0 0
\(851\) −7.95930 −0.272841
\(852\) 0 0
\(853\) −44.5599 −1.52570 −0.762851 0.646575i \(-0.776201\pi\)
−0.762851 + 0.646575i \(0.776201\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −42.4400 −1.44972 −0.724861 0.688895i \(-0.758096\pi\)
−0.724861 + 0.688895i \(0.758096\pi\)
\(858\) 0 0
\(859\) 56.9998 1.94481 0.972405 0.233300i \(-0.0749524\pi\)
0.972405 + 0.233300i \(0.0749524\pi\)
\(860\) 0 0
\(861\) 37.0975 1.26428
\(862\) 0 0
\(863\) −46.0210 −1.56657 −0.783287 0.621660i \(-0.786459\pi\)
−0.783287 + 0.621660i \(0.786459\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 2.84431 0.0965979
\(868\) 0 0
\(869\) −14.2029 −0.481799
\(870\) 0 0
\(871\) −6.72205 −0.227768
\(872\) 0 0
\(873\) −26.7934 −0.906820
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −45.5475 −1.53803 −0.769015 0.639231i \(-0.779253\pi\)
−0.769015 + 0.639231i \(0.779253\pi\)
\(878\) 0 0
\(879\) −84.1715 −2.83903
\(880\) 0 0
\(881\) 41.0283 1.38228 0.691140 0.722721i \(-0.257109\pi\)
0.691140 + 0.722721i \(0.257109\pi\)
\(882\) 0 0
\(883\) 3.67625 0.123716 0.0618578 0.998085i \(-0.480297\pi\)
0.0618578 + 0.998085i \(0.480297\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −9.00591 −0.302389 −0.151194 0.988504i \(-0.548312\pi\)
−0.151194 + 0.988504i \(0.548312\pi\)
\(888\) 0 0
\(889\) 32.9455 1.10496
\(890\) 0 0
\(891\) −10.1840 −0.341177
\(892\) 0 0
\(893\) 9.08650 0.304068
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −7.59208 −0.253492
\(898\) 0 0
\(899\) −31.5178 −1.05118
\(900\) 0 0
\(901\) −31.0975 −1.03601
\(902\) 0 0
\(903\) −59.4907 −1.97973
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 14.5687 0.483746 0.241873 0.970308i \(-0.422238\pi\)
0.241873 + 0.970308i \(0.422238\pi\)
\(908\) 0 0
\(909\) −0.530184 −0.0175851
\(910\) 0 0
\(911\) 18.3694 0.608605 0.304303 0.952575i \(-0.401577\pi\)
0.304303 + 0.952575i \(0.401577\pi\)
\(912\) 0 0
\(913\) −18.5833 −0.615016
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 15.3259 0.506107
\(918\) 0 0
\(919\) 26.6130 0.877881 0.438940 0.898516i \(-0.355354\pi\)
0.438940 + 0.898516i \(0.355354\pi\)
\(920\) 0 0
\(921\) −69.3191 −2.28414
\(922\) 0 0
\(923\) 0.690967 0.0227435
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 43.7738 1.43772
\(928\) 0 0
\(929\) 13.2939 0.436159 0.218079 0.975931i \(-0.430021\pi\)
0.218079 + 0.975931i \(0.430021\pi\)
\(930\) 0 0
\(931\) 0.901116 0.0295329
\(932\) 0 0
\(933\) −57.6894 −1.88867
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −45.2980 −1.47982 −0.739911 0.672705i \(-0.765132\pi\)
−0.739911 + 0.672705i \(0.765132\pi\)
\(938\) 0 0
\(939\) 59.4362 1.93963
\(940\) 0 0
\(941\) 41.5192 1.35349 0.676743 0.736219i \(-0.263391\pi\)
0.676743 + 0.736219i \(0.263391\pi\)
\(942\) 0 0
\(943\) −16.8282 −0.548002
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −5.65658 −0.183814 −0.0919071 0.995768i \(-0.529296\pi\)
−0.0919071 + 0.995768i \(0.529296\pi\)
\(948\) 0 0
\(949\) −0.778849 −0.0252825
\(950\) 0 0
\(951\) −72.9505 −2.36558
\(952\) 0 0
\(953\) −33.2967 −1.07858 −0.539292 0.842119i \(-0.681308\pi\)
−0.539292 + 0.842119i \(0.681308\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −28.7280 −0.928643
\(958\) 0 0
\(959\) 41.8158 1.35030
\(960\) 0 0
\(961\) −10.6552 −0.343716
\(962\) 0 0
\(963\) −40.4079 −1.30213
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −36.4647 −1.17263 −0.586313 0.810084i \(-0.699421\pi\)
−0.586313 + 0.810084i \(0.699421\pi\)
\(968\) 0 0
\(969\) −10.3214 −0.331572
\(970\) 0 0
\(971\) 22.9047 0.735046 0.367523 0.930014i \(-0.380206\pi\)
0.367523 + 0.930014i \(0.380206\pi\)
\(972\) 0 0
\(973\) 61.6413 1.97613
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 6.57736 0.210428 0.105214 0.994450i \(-0.466447\pi\)
0.105214 + 0.994450i \(0.466447\pi\)
\(978\) 0 0
\(979\) −8.37959 −0.267813
\(980\) 0 0
\(981\) 57.9912 1.85152
\(982\) 0 0
\(983\) −25.4451 −0.811571 −0.405786 0.913968i \(-0.633002\pi\)
−0.405786 + 0.913968i \(0.633002\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 66.1099 2.10430
\(988\) 0 0
\(989\) 26.9862 0.858113
\(990\) 0 0
\(991\) 41.5090 1.31858 0.659288 0.751890i \(-0.270858\pi\)
0.659288 + 0.751890i \(0.270858\pi\)
\(992\) 0 0
\(993\) −49.6232 −1.57474
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 15.4486 0.489263 0.244631 0.969616i \(-0.421333\pi\)
0.244631 + 0.969616i \(0.421333\pi\)
\(998\) 0 0
\(999\) −4.36721 −0.138173
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 1900.2.a.g.1.1 3
4.3 odd 2 7600.2.a.ca.1.3 3
5.2 odd 4 1900.2.c.f.1749.6 6
5.3 odd 4 1900.2.c.f.1749.1 6
5.4 even 2 1900.2.a.i.1.3 yes 3
20.19 odd 2 7600.2.a.bl.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1900.2.a.g.1.1 3 1.1 even 1 trivial
1900.2.a.i.1.3 yes 3 5.4 even 2
1900.2.c.f.1749.1 6 5.3 odd 4
1900.2.c.f.1749.6 6 5.2 odd 4
7600.2.a.bl.1.1 3 20.19 odd 2
7600.2.a.ca.1.3 3 4.3 odd 2